Preheat oven to 375°. In a medium bowl, combine 1 cup cheddar, mozzarella, cream cheese, Dijon mustard, chives, garlic powder, beer, and ¼ tsp each salt and pepper. Mix until completely combined.
Roll out biscuit dough on a lightly floured surface to about ⅛-inch thick. Cut out circles of dough with a biscuit cutter or the rim of a glass and transfer to a parchment-lined baking sheet.
Brush the top of each circle with egg wash. Combine baking soda and remaining ¾ cup cheddar in a small bowl, then sprinkle over top of each biscuit. Sprinkle with coarse salt.
Bake until golden brown and cooked through about 10 minutes. Let cool slightly before serving warm with Pretzel Ring Beer Cheese Dip.