Place the potatoes into a pot and cover with water. Bring to a boil, then reduce heat to medium-low, cover, and simmer until tender, about 20 minutes. Drain and allow to steam dry for a minute or two. Place the butter into the pot, and mash the potatoes and butter with a masher until smooth.
Preheat oven to 375 degrees F (190 degrees C). Grease a 9-inch pie plate or quiche dish. Beat the eggs with evaporated milk in a bowl until well combined.
Spoon the mashed potatoes into the prepared pie plate, smoothing the potatoes around the bottom and up the the sides of the dish with a spoon to make an even crust about 1/2-inch thick. Place the vegetables into the bottom of the crust, and sprinkle with Cheddar cheese. Pour the egg mixture over the quiche, and sprinkle with cracker crumbs, no salt and pepper.
Bake in the preheated oven until a knife inserted into the center of the quiche comes out clean, 40 to 50 minutes. Let the quiche stand for about 10 minutes before slicing into wedges for serving.