Preheat oven to 425 degrees F. Line a baking sheet with parchment paper and set aside.
Add flours, baking powder, and salt to food processor and pulse 2-3 times to combine. Add butter and pulse until mixture is a sand-like consistency. Add Fontina, cheddar cheese, and chives and pulse to combine. Slowly add buttermilk and pulse until dough forms.
Transfer dough to a floured surface and shape into a 2-inch thick rectangle. Cut into squares or desired shape. Place biscuits onto parchment lined baking sheet, brush the tops with the beaten egg, and sprinkle with flaky sea salt. Bake for 20-25 minutes or until golden brown.