There are two types of ice makers, modular and component models. Both allow you to test and replace the thermostat individually. To determine which model you have remove the outer face plate. If there is knob, pull it off first, then pry off the face plate with a small screwdriver.
It is easier to work on this type of unit by first removing the entire ice maker from the freezer compartment. The unit is usually mounted to the freezer case by two or more screws. Remove the bottom screws first. While supporting the ice maker, remove the remaining top screws. Unplug the ice maker from the freezer cabinet if you have not already done so.
The inner mounting plate is held in place by two or more screws. Near the edge where the shutoff arm is located, a pin (labeled "A" below) protrudes through the plate. This pin has a spring under the mounting plate. Reduce the tension on the spring by moving the shutoff arm down and next to the ice maker. Remove the mounting plate.