Rinse shiitake and common mushrooms. Trim and discard shiitake stems; coarsely chop both kinds of mushrooms (keep separate). In a 10-inch frying pan over medium-high heat, melt 1 tablespoon butter with 1 tablespoon olive oil. Add shiitakes and stir often until browned, 4 to 5 minutes. Pour into a bowl. Melt another tablespoon butter with another tablespoon olive oil in pan. Add common mushrooms and stir often over medium-high heat until liquid evaporates and mushrooms are well browned, 10 to 12 minutes. Add to bowl. Cool slightly, then finely chop mushrooms.
In same pan, over medium-high heat, melt third tablespoon butter. Add onion and stir often until beginning to brown, about 8 minutes. Add garlic and stir until fragrant, about 1 minute. Pour in vinegar and stir until liquid evaporates, 1 to 2 minutes. Stir in parsley and the mushrooms; season to taste with salt and pepper.
Combine flour and 1/2 teaspoon salt in the bowl of a food processor. Whirl briefly to mix. Add 6 tablespoons chilled butter and whirl until coarse crumbs form. Pour in 1/4 cup cold water and whirl just until dough comes together. Squeeze a little in your hand; if it doesn't hold together, add another tablespoon or two of water and pulse to mix. Pat dough into a ball, then flatten slightly into a disk, wrap in plastic, and chill until cold, 15 to 20 minutes.
Preheat oven to 375°. On a well-floured surface, with a floured rolling pin, roll dough into an oval 1/4 to 1/8 inch thick. Brush half with beaten egg. Spoon tablespoonfuls of the mushroom mixture into one hand, squeeze tightly, and mound an inch apart on egg-washed side of dough. Fold bare half over mounds, and press between them to seal. With a 2-inch cookie cutter, cut out tarts and transfer to a baking sheet. Gather remaining dough, roll again, fill, and cut out more tarts.
Bake until beginning to brown, about 15 minutes. Brush tarts with egg wash (discard any remaining) and continue baking until deep golden brown, about 5 minutes longer. Serve warm.
Wine pairing: Pick a Pinot Noir with earthy loam and mushroom notes to echo the mushrooms in the tarts.