Melt the butter in a large saucepan.

Add the sliced onions. Cook over low heat, tossing occasionally with a spatula, for 1 hour or until onions turn soft, sweet, and golden brown. Set aside to cool.

Preheat oven to 400° F (205° C). Line a half-sheet rimmed tray with baking parchment.

On a silicone rolling mat or lightly floured countertop, unroll puff pastry. With a rolling pin, roll the pastry a few inches wider, or a sharp knife to trim away any excess. When finished, you should have a 9 x 13-inch rectangle. Position it with the longer side facing you.

Slice the rectangle in half horizontally to form two 4 1/2 x 13-inch rectangles.

Transfer one rectangle to the prepared tray.

Use a lattice cutter to cut the second rectangle.

With your fingertips, gently pull the cut pastry apart to reveal the open weave pattern.

Scatter the un-cut rectangle with the grated cheddar cheese, allowing a 3/4-inch border along the two longer sides.

Arrange the caramelized onions in a mound on top of the cheese, distributing them evenly along the length of the pastry. Season with salt and pepper.

With a pastry brush, brush the beaten egg around the border of the puff pastry.

Place the lattice-cut pastry over top of the cheese and onion mixture. With your fingertips, pat down the pastry until it fits snugly around the cheese and onion mixture. Trim away the excess pastry with a sharp knife, creating clean, even edges.

With a fork, crimp the edges of the pastry to seal in the filling.

Brush the top of the pastry with the beaten egg.

Bake in preheated oven, about 30 minutes, or until flaky and deeply golden brown.

With a serrated knife, slice the pastry into 1 1/2 – 2-inch wide slices. Serve warm.