Preheat the oven to 400 degrees F. Combine 4 cups of flour, the baking powder, and salt in the bowl of an electric mixer fitted with a paddle attachment.
Add the butter and mix on low speed until the butter is in pea-sized pieces. Mix the eggs and heavy cream and quickly add them to the flour-and-butter mixture.
Combine until just blended. Toss together the Cheddar, dill, and 1 tablespoon of flour and add them to the dough.
Mix until they are almost incorporated. Dump the dough onto a well-floured surface and knead it for 1 minute, until the Cheddar and dill are well distributed. Roll the dough 3/4-inch thick.
Cut into 4-inch squares and then in half diagonally to make triangles. Brush the tops with egg wash. Bake on a baking sheet lined with parchment paper for 20 to 25 minutes, until the outside is crusty and the inside is fully baked.