Combine flour, sugar, baking powder, and salt in a large mixing bowl. Use a pastry cutter to cut butter into flour mixture until it resembles rice or coarse meal. Stir in ham and Gruyère until evenly mixed.

Moisten flour mixture with cream until dough begins to come together. Add a bit more cream, milk, or water if dough seems dry.

Turn out on a floured work surface and fold dough onto itself. Gently stretch dough, turning and folding 3 or 4 times. Lightly flour a rolling pin and roll dough out 1/2- to 3/4-inch thick, turning dough a quarter turn occasionally.

Cut scones with a lightly floured 3-inch biscuit cutter; press cutter straight down, without twisting. Pat dough scraps together and cut as needed. Line a baking sheet with parchment; place cut scones on the sheet.

Whisk egg, water, and pinch of salt together in a small bowl. Brush tops of scones with egg wash, and refrigerate for about 20 minutes.

Preheat the oven to 450 degrees F (235 degrees C). Brush the top of the scones with the egg mixture again.