Place mozzarella cheese and cream cheese in a microwave-safe bowl. Microwave on high, 20 seconds at a time, until melted.

Whisk egg in a bowl until beaten. Mix in almond flour, baking powder, and garlic powder until well combined. Work mozzarella mixture into the dough until sticky. Stir in Cheddar cheese.

Transfer the dough to a sheet of plastic wrap and fold plastic wrap over the dough. Gently work the dough into a ball. Refrigerate for 30 minutes.

Preheat the oven to 425 degrees F (220 degrees C). Grease a baking sheet or line with parchment paper.

Remove the dough from the refrigerator and unwrap. Cut dough into 4 equal pieces and roll each piece into a ball. Cut each ball in half to form a top and bottom bun. Place dough, cut-side down, on the prepared baking sheet.

Bake in the preheated oven until golden and set up, 10 to 12 minutes.