In a large bowl, whisk together flour, sugar, baking powder, salt, and pepper. Add cold butter and Gruyère, tossing to coat. Cover and freeze for 15 minutes. Stir in ham and thyme.

In a small bowl, whisk together milk and 1 egg. Add milk mixture to flour mixture, stirring and folding just until moistened.

Turn out dough onto a heavily floured surface. Using well-floured hands, knead dough 5 to 10 times, about 1 minute. Shape dough into an 8-inch circle. Using a bench scraper or sharp knife, cut into 8 wedges. Place about 1½ inches apart on prepared pan.

In a small bowl, whisk together cream and remaining 1 egg. Brush egg wash onto scones. Sprinkle with flake sea salt.

Bake until golden brown, 15 to 20 minutes. Let cool slightly; serve warm.