Preheat the oven to 400˚. Roll out the puff pastry into a 10-inch square on a lightly floured surface. Cut into 8 strips, then cut each in half crosswise. Pat the sausages dry with a paper towel and cut each in half crosswise. Wrap a strip of puff pastry around each sausage, gently rolling the seams against a clean surface to seal.
Arrange the wrapped sausages on a baking sheet, leaving a little space between each. Brush with the beaten egg and sprinkle with the cheese and 3/4 teaspoon pepper. Bake until golden, 25 to 30 minutes.
Combine the mayonnaise, sour cream, mustard and garlic in a small bowl; season with salt and pepper. Serve with the sausages.