In a medium pot over medium heat, melt butter. Add shallot and cook, stirring occasionally, until softened and translucent, 3 to 4 minutes. Whisk in flour to form a thick, bubbling paste, then whisk in milk. Increase heat to medium-high and bring to a boil, then reduce heat to medium low and simmer, stirring occasionally, until thickened, 1 to 2 minutes.

Remove pot from heat and whisk in 1 cup Gruyère just until melted; season with salt and pepper. Fold in spinach and 1 cup ham. Let cool.

Line 2 baking sheets with parchment. On a lightly floured surface, roll 1 sheet of puff pastry to a 15"-by-10" rectangle. Using a sharp knife, cut into 6 equal squares, each about 5"x5". Transfer squares to one of the prepared sheets. Repeat with remaining sheet of puff pastry and baking sheet.

Dollop 3 tablespoons cooled filling into center of each square. Press 5 to 6 pieces of remaining ham into top of each. Sprinkle with remaining ½ cup Gruyère. Fold corners of each square to meet in the center. Refrigerate pastries until chilled, 15 to 30 minutes.

Arrange racks in upper and lower thirds of oven; preheat to 400°. In a small bowl, whisk egg, water, and a pinch of salt. Brush egg wash over chilled pastries.

Bake puffs, rotating sheets top to bottom after 10 minutes, until pastry is golden brown on top and bottom, 15 to 20 minutes. Top with more pepper and serve warm.