Heat oven to 425°F. Add flour, baking powder, and salt to the bowl of a stand mixer fitted with the paddle attachment. Mix to combine, then add butter and mix on low speed until dough resembles coarse crumbs.

Beat 1 egg with 1 tablespoon of milk in a small bowl. Add buttermilk and whisk to combine. Add buttermilk mixture to the dough and mix on low speed until moistened.

Add cheddar and a handful of flour to a small bowl and stir to combine. Add cheese mixture to the dough and mix on low speed until combined.

Line a sheet pan with parchment paper. Beat the second egg in a small bowl with a bit of water to create an egg wash. Set aside.

Knead dough lightly on a well-floured board, making sure to not overwork it. Roll dough out to a roughly 10-inch-by-5-inch rectangle. Form ¼ cup-size portions and place onto the sheet pan. Brush tops with egg wash and bake for 20 to 25 minutes. Serve warm.