Reddit reviews HiLetgo DC 3-5V MAX6675 Module + K Type Thermocouple Temperature Sensor Thermocouple Sensor Set M6 Screw for Arduino

The Maxim chips look to be the way to go, in one IC they are doing all the measurement and signal filtering to give you digital data. The Maxim 31855 is the [SPI example at Adafruit] (https://learn.adafruit.com/circuitpython-basics-i2c-and-spi/spi-devices). That guide walks through how they retrieve the data over SPI then use the datasheet to lookup the temperature. Past that there are libraries for the [8155 and 6675] (https://github.com/adafruit?q=MAX) which is [$7 on Amazon] (https://www.amazon.com/HiLetgo-MAX6675-Thermocouple-Temperature-Arduino/dp/B01HT871SO).
Not familiar with the 2560 but looks like you would use pins 50-52.

I'm going to assume you know what you are doing on the reflow, profiles and layers well out of my depth.
The short, and mostly correct, answer is yes the wires running through the oven will have you reading the oven temperature.
For how a thermocouple works there is a good writeup on [brighthubengineering.com] (http://www.brighthubengineering.com/manufacturing-technology/53682-what-is-a-thermocouple-how-thermocouple-works/). The Thomson effect is the one that allows you to measure the temperature of the mobo with a long probe.
I can't think of a way for you to use a thermocouple to measure the temperature of something actually inside the oven. They can be used to measure temperatures where the cold junction is of a higher temperature than the hot junction so there may be a way I don't understand.

https://www.amazon.com/Yosoo-5V-12V-Voltage-Induction-Heating/dp/B01C71XKZ6

​

That's the one I'm using. Along with this MAX6675 thermocouple module to measure temperature:

​

https://www.amazon.com/HiLetgo-MAX6675-Thermocouple-Temperature-Arduino/dp/B01HT871SO/ref=sr_1_2?ie=UTF8&qid=1539837568&sr=8-2&keywords=thermocouple+arduino