Preheat the oven to 375 degrees F (190 degrees C).
Lay bagel halves with their centers facing up on a baking sheet. Use a sharp knife to gently score around the inside of the bagel (close to the very center and along the outside edge). Remove the bready centers, being careful not to rip through the bagel. Discard the middle pieces of bread, or save for bread crumbs or another use.
Gently press all the way around the middle and sides with your fingers to create a well around the center of each bagel half.
Whisk ham, egg, half-and-half, shredded Cheddar, salt, and pepper together in a bowl until combined. Divide the egg mixture between the prepared bagels.
Bake in the preheated oven until bagels are well toasted and egg is set in the middle, 12 to 15 minutes. Cool briefly before serving.