Cut the beef into 1-inch cubes and season it with 1 teaspoon salt. Spread the pieces in one layer on a baking sheet and freeze until solid, at least 1 hour.

Take the puff pastry out of the freezer at least 2 hours before using it, following the package directions for thawing.

Melt the butter in a large nonstick or cast-iron skillet over medium-low heat.

Pulse the shallot and garlic in a mini food processor for about 30 seconds, until finely minced. Add them to the skillet and cook, stirring often, for about 3 minutes, until soft.

Pulse the mushrooms in two batches until finely minced and add them to the skillet with 1/4 teaspoon salt and 1/4 teaspoon pepper.

Cook the mushrooms over medium heat, stirring now and then, for 17 to 20 minutes until soft and just starting to brown.

Add the red wine and cook for about 2 minutes until it evaporates. Remove the pan from the heat, stir in the sage, and transfer the mixture to a bowl to cool.

In a separate bowl, stir the softened Boursin and Dijon together.

Unfold the pastry and cut each sheet along its seams into three strips. Roll each strip to 15 x 3 inches, then cut it into five squares.

Place 1 teaspoon of mushroom mixture on the bottom third of each square, leaving 1/4 inch clear at the bottom. Add 1/2 teaspoon of the cheese mixture just above it, saving the rest for the dipping sauce. Set one frozen beef cube between the two fillings.

Beat the egg with 1 teaspoon water. Brush the egg wash along the edges of each square.

Fold the pastry over the filling, press to seal, crimp the edges with a fork, and trim any excess pastry.

Arrange the bites on a parchment-lined baking sheet, brush the tops with egg wash, and bake for 10 to 12 minutes until puffed and golden.

Whisk cream, half-and-half, or milk into the reserved cheese mixture until it becomes a smooth dipping sauce.

Serve the sauce with the warm bites.