Preheat the oven to 425 degrees F. Line a rimmed baking sheet with parchment paper. Cook the sausage in a medium nonstick skillet over medium-high heat, stirring, until browned, 4 to 6 minutes; set aside to cool.
Unfold the puff pastry on the prepared baking sheet; spread the mustard on top, leaving a 1-inch border all around. Spread the sausage on one half of the pastry and top with the cheese curds. Fold the other half of the pastry over the cheese to cover; crimp the edges with a fork to seal. Freeze until firm, about 45 minutes.
Brush the pastry all over with the beaten egg and sprinkle with flaky salt. Cut 4 slits in the top to let steam escape. Bake until the pastry is golden brown and puffed all over, about 30 minutes. Let cool at least 20 minutes, then transfer to a cutting board to slice. Serve warm or at room temperature with more mustard on the side.