Place the thawed puff pastry sheet on a large cutting board and lightly dust the surface with a little flour using your hands so the pastry does not stick when you cut it.​

Use a chef’s knife to cut the puff pastry into 12 equal squares by first cutting it into 3 strips, then cutting each strip into 4 pieces, and set the squares aside on the board.​

Lightly grease a 12-cup muffin tin with a small amount of olive oil using a pastry brush or a paper towel so the tartlets release easily after baking.​

Lay the asparagus on the cutting board, trim off the woody ends with the chef’s knife, then cut the spears into 1-inch pieces so they fit inside the muffin cups.​

Place the asparagus pieces into a small mixing bowl, drizzle 1 tbsp olive oil over them with a measuring spoon, sprinkle ¼ tsp kosher salt and ⅛ tsp black pepper, and toss with your hands or a spoon until coated.​

Put the Brie on the cutting board, remove the rind with the chef’s knife if you prefer a smoother melt, then cut the cheese into 12 small cubes, one for each tartlet.​

Take one puff pastry square and gently press it down into a muffin cup with your fingers, letting the corners rise up the sides to form a little pastry cup; repeat with the remaining squares until all 12 cups are lined.​

Use a fork to poke the bottom of each pastry cup 2 or 3 times, creating small holes that help prevent the pastry from puffing too high in the center.​

Drop one cube of Brie into the center of each puff pastry cup with your fingers, pressing it lightly so it sits flat at the bottom.

Spoon a small handful of asparagus pieces into each cup on top of the Brie using a regular spoon, filling each tartlet almost to the top but not overflowing.

Sprinkle grated Parmesan over the top of each tartlet with a measuring spoon so a light layer of cheese covers the asparagus; this Parmesan melts and crisps into frico-style golden edges around the pastry and filling.​

Crack 1 egg into a small bowl, beat it with a fork until smooth, then use a pastry brush to lightly brush the exposed puff pastry edges on each tartlet with egg wash so they turn shiny and golden in the oven.​

Place the muffin tin on the middle rack of a preheated 400°F oven using oven mitts, and bake for 18 minutes until the puff pastry is well risen and deep golden, the Brie has melted, and the Parmesan on top has crisped and turned light brown around the edges.​

Remove the muffin tin from the oven with oven mitts and set it on a heat-safe surface, then let the tartlets cool in the pan for 5 minutes so the cheese sets slightly and the pastry firms up.

Use a small knife or the edge of a fork to gently loosen each tartlet from the muffin cup, then lift them out and transfer to a serving plate with your hands or tongs, and serve warm as an Italian-style appetizer.