What you need are a couple of things:
1) Something to switch 115V at a current of around 2A. Since the device contains a motor, it may be prudent to ensure it can switch a heavier load (so let's say 5A).
2) a method of producing a variable time delay
3) a power supply.
4) a method to detect that a request to turn the power off.
Another suggestion, knowing the actual requirements, is that you probably want to have some difference in the trip point so that you're not alternately filling and emptying as the temperature hovers around 34F. I'll assume it actually has some hysteresis (the temperature needs to rise a few degrees above the temperature it turned off before it turns back on again).
(1) switching the load.
The most critical thing is (1), and there are several options.
a) a mechanical relay
b) a solid state relay
c) a triac
d) another method.
You would be familiar with mechanical relays I assume. A solid state relay is similar in nature, but has no moving parts, operating silently. Both are isolated, meaning that the contacts used to energise the external circuit are electrically isolated from those of the external circuit.
In contrast, a triac is not isolated. The control connections require a connection to the load. There are methods of isolating the control circuitry from the load, but these require several components to be "hot". You can actually think of a solid state relay to be something like a triac and the isolating circuitry all contained in a box so that the external connections are isolated.
A triac is not a transistor. A 2N3055 is a power transistor. Especially important is the maximum voltage of the transistor. If exposed to more than this is will go into a form of conduction and usually destroy itself. FOr the 2N3055, that voltage is around 60V. The peak voltage of 115V AC is 160V, so the 2N3055 is already ruled out. In addition, the 2N3055 can't switch the AC voltage as it goes from positive to negative. At best it could switch only half cycles in one direction. At worst, the negative half cycles could destroy it. (sure, there are ways of getting past this, but again, the limit on maximum voltage is the main problem, and anyway the "correct" part -- a triac -- is going to be less expensive.)
There are other semiconductor devices that can be used, but they're going to have similar features to either/both a triac and/or a transistor (in fact many solid state relays (SSRs) use one of these other devices.
(2) the variable delay.
Now, from your better explanation above, I assume this is for the blower. And I also assume that it needs to turn on *as* the thermostat switches off, but only stays on for ~60 seconds. Furthermore, if the thermostat cycles back on during the 60 seconds (undesirable as it might be) I would assume you would want it to turn off the blower immediately.
The action you're describing here is a form of monostable where the action of the power going off triggers the timer to turn something on for a variable (~60 second) period. The triggering needs to be such that the monostable is immediately turned off if the power comes back on -- this is a little unusual.
There is a common chip that can do all sorts of timing tasks. It is the 555. Variable delays ranging from very small fractions of a second to the minute that you require are viable, and a circuit to provide a variable delay of a couple of seconds to about a minute can be fairly easily developed.
This circuit will operate from a low DC voltage (maybe between 6 and 15 volts) and will be capable of operating an appropriate relay, SSR, or the circuit that isolates a triac. However I'd go for one of the former 2 options (relay or SSR).
The 555 circuit will need to be built up on a board (there are various options available) and placed in a box along with the relay. The circuit will require about 10 other components -- capacitors, resistors, and the variable resistor to set the delay.
(3) A power supply.
Because the circuit needs to be powered, you need a power supply. This might be batteries, or it might be a small plugpack (wall wart).
Note that this does NOT get switched on and off. The circuit needs to be powered while the thermostat has turned OFF. (OK, it could be powered ONLY while the thermostat is off -- simplifying some of the issues)
(4) A method to detect the triggering signal.
If the device is only powered whilst the thermostat is turned off, it simply needs to turn the load on immediately then turn it off after a delay -- resetting if the power to it is removed (which will also turn off the power to the load)
If not powered this way, you need to detect the thermostat has gone off (and subsequently back on). This may be through a dedicated pair of contacts, or even via another power supply which produces a low DC voltage while the thermostat is "off".
As you can see, there are quite a few things required. All of these things (or equivalents) are inside a time delay relay.
Yes you can build something like this, and maybe the live electrical stuff would not be a challenge for you. However, the cost might well be higher than a time delay relay. This is especially a concern since the required function is probably no greater than what a commercially purchased time delay relay can provide.
Sure, if you want to build up your own circuit to do this we can help you, but you need to weigh up whether you're doing this for pure function, or whether there is something additional you'll get out of it that will justify the increased cost and complexity.
