Arrange an oven rack in the middle position and preheat the oven to 400 degrees. Line a rimmed baking sheet with parchment paper or a silicone mat.
In a large saute pan, melt butter until foaming. Add leeks and mushrooms and saute until tender, about 10 minutes. Stir in the garlic until fragrant, about 30 seconds.
Add 1 teaspoon of salt, 1/4 teaspoon of freshly ground pepper, Havarti cheese, thyme, basil, and the heavy cream. Bring the mixture to a boil and stir until the mixture has thickened, about 2 minutes. Season to taste with salt and pepper. Remove from heat and cool slightly.
Unfold both sheets of puff pastry and place on prepared baking sheet, leaving space between them. Spoon half the cheese mixture onto the middle of each sheet. Fold the edges over to form the rim of the tart.
In a small bowl, whisk the egg and water together to create an egg wash. Using a pastry brush, brush the edges of the each of the tarts. Divide the Parmesan between the tarts, sprinkling it over the top.
Bake until the pastry is deep golden brown in color, approximately 20 to 25 minutes. Cool 5 minutes. Serve warm or at room temperature.