Hmmh, just to point out that something like this can be done, about 15 years ago there was an article in a magazine (perhaps Elektor Electronics) of a DIY project to connect a multimeter to a PC. It was done with a small PCB that attached to the multimeter's LCD pins. It had a microcontroller which snooped the LCD signals and tracked which numbers were on the screen and then sent the numbers to a PC via a serial port.
You might be able do the same with this LED display. You'll find the pinout for that LED module here: http://www.aplomb.nl/Niels_skn/7-Segment_QRD%20Niels.pdf. There are 4 commons (pins 6, 8, 9, 12) and 8 segments. For each common, detect when it's pulled to the ground. Then measure the 8 segment voltages to find out what is the number in that position.
Before starting to build anything, measure them with an oscilloscope to confirm what the voltage levels are. If you remove the display and instead put pull-up resistors to the commons and pull-down resistors to the segments, the voltage levels will probably be easier to measure. In any case, I wouldn't connect anything directly to the pins, I'd use a series resistor (10K) in any case. This is also true regarding the ground. Try to figure out how the ground is connected so that there are no surprises, like, if the device's "ground" is not at 0 volts.
Whatever you do, be careful! There might be serious safety concerns in your particular application as it turns out you're modifying some kind of a stove.