Combine mozzarella cheese and cream cheese in a large microwave-safe bowl. Heat in microwave until slightly melted, about 45 seconds. Remove from microwave and stir. Repeat, heating for about 20 seconds more. Remove from microwave again and stir; consistency should be very sticky and cheese should be completely melted. Return to microwave, if necessary, until all cheese is melted.
Mix flour, eggs, chives, baking powder, and garlic powder together in a bowl. Add mixture to large bowl with melted cheeses and mix well. Add Cheddar cheese and stir into dough.
Lay a sheet of plastic wrap on a flat surface and dust lightly with flour. Pour dough onto middle of plastic wrap and roll up to form a ball. Place in refrigerator until slightly hardened, 20 to 30 minutes.
Preheat oven to 425 degrees F. Grease a baking sheet with butter.
Remove dough from refrigerator and cut into 8 equal-sized pieces. Roll each into a ball and place on prepared baking sheet, spacing equally, as biscuits will spread while cooking.
Bake in preheated oven until golden brown, checking at 10 minutes. Remove from oven immediately and cool completely before serving, about 15 minutes.