Butter the inside of a 6-quart slow cooker. Pat the thawed potatoes dry and mix them with the kielbasa, pepper and onion.
Put half the potato mixture and half the cheese in the cooker, then repeat the layers.
Whisk the eggs, milk, salt and pepper. Pour over the layers and press lightly to settle the filling.
Cover and cook on LOW for 6 to 7 hours, beginning temperature checks at 5 1/2 hours. Stop when the center is set and reaches 165°F; do not use a delayed start with raw eggs.
Turn off the cooker and remove the lid for 10 minutes before serving with a broad spoon.