Preheat oven to 375° F.
Unfold the sheet of puff pastry. Using a sharp knife, trim off the corners to make a round about 12 inches in diameter. Lay the round on a sheet of parchment paper and transfer to a baking sheet. Pinch the edges up to form a generous ¼-inch rim. In a bowl, combine the apples, 2 tablespoons of the sugar, and lemon juice and toss well. Spread the apricot jam on the pastry and place in the freezer for 10 minutes. Sprinkle with the grated cheese, then arrange the apple slices in concentric circles on top. Sprinkle the apples with the remaining sugar. Brush the edge of the pastry with the milk. Return to the freezer for 10 minutes.
Bake until the pastry is puffed and browned and the apples are lightly golden, about 30 minutes. Remove from the oven, cover loosely with foil and let stand 5 minutes. Use a long knife, such as a carving knife, to separate the galette from the parchment paper and remove the galette to a cake plate.
To serve, cut into wedges.