Heat butter in a skillet over medium heat. Add garlic and shallots and sauté for 1 minute. Add Dijon and stir to combine. Reduce heat to low.
Add cream cheese and stir until melted. Add grated cheddar and stir until melted. Turn off heat. Add salt and pepper to taste. Fold in egg whites.
Dunk bread cubes in cheese, coating thoroughly. Place on a nonstick baking mat or sheet of waxed paper and freeze for 20 minutes, uncovered. Remove frozen chunks from tray and place into a Ziploc bag. Store in freezer.
When ready to bake, preheat oven to 375 degrees. Place frozen cheese puffs on a baking sheet with a nonstick baking mat or parchment paper and bake for 10 minutes or until golden brown. Serve immediately or at room temperature.