Designing a dental furnace control board
Buy an off-the-shelf PID self-tuning temperature controller. Find out what the sensor used in the oven is first. You can use the controller to switch a mechanical contactor, a mercury displacement relay or an SSR. Make sure the controller matches the output device as well. Output can be a relatively long 'PWM' (a few seconds if a zero-voltage switching SSR or about 20 seconds if a mechanical contactor). If you use an SSR you will have about 8-10W to get rid of so you may need a big heatsink or a smaller heatsink and a fan. 

The controllers are not particularly expensive, a few hundred dollars for a really good one, and crappy ones from China are almost free (but tend to be only type K, still probably better than something you would build on the first try).  

Some have optional communications via RS485, for example. If you need to get it to talk to something, buy one with communications and you can hook it to a PC or an Arudino or a RPi etc.  Some have programmable ramp and soak. 

The sensor is undoubtedly a thermocouple, probably type K, N, S or R. The latter two are platinum alloy types, the first are base metal alloy types and will be much thicker wire at the sensing junction. You must maintain thermocouple wire (or proper extension leadwire), in the correct polarity, from the sensor right back to the controller terminal block. The Pt alloy sensors are very low output per degree so a better controller is indicated to maintain accuracy. 

Keep the electronics cool and it will last a very long time. 



If you really, really want to spend the time to develop a controller, which is not remotely a cost-effective use of time unless you intend to sell these in fair quantities, first you must figure out how to measure the sensor, including signal conditioning, cold-junction compensation and linearization, with acceptable noise level (more challenging for Platinum types) and update rate (10x per second is more than enough, less is probably acceptable. You can buy chips to do some or all of this, with varying degrees of performance and limitations. 

Then you must implement a PID algorithm for control. There is plenty of information out on the web on PID- if you care about overshoot and good control it may be non-trivial. 

After those two, you can proceed as if you had purchased a simple controller. Otherwise, implement timing and ramping for a more complex programmable controller.