Line a baking sheet with parchment paper, then set aside.
In a medium bowl, stir to combine flour, sugar, baking powder, garlic powder, black pepper, salt, and paprika. Add butter and work in with your fingertips or a pastry cutter until only pea-sized bits remain visible.
Fold in the cheese, ham, and scallions.
Whisk to combine ⅔ cup of half-and-half with the egg. Pour into dry mixture and toss lightly to combine. Pour the shaggy dough onto the prepared baking sheet and gently press to form a 4x14-inch rectangle (it should be about 1 ½-inches thick). Refrigerate for 30 minutes.
While the dough is chilling, preheat the oven to 400 F.
Cut dough into 5 equal sections. Then cut each section into 2 triangles. Space the triangles out across the baking sheet. Brush the tops with the remaining half-and-half and sprinkle lightly with coarse salt, if using.
Bake for 24-26 minutes until golden brown. Transfer to a rack to cool slightly and serve warm.