Preheat your oven to 400°F. Line a baking sheet with parchment paper.
Toss the mushrooms, shallots, garlic, olive oil, and balsamic vinegar directly on the baking sheet. Season with salt and pepper before spreading out into an even layer. Roast for 20-22 minutes, stirring midway, until the mushrooms are golden and tender.
Increase the oven temperature to 425°F. Arrange the flatbreads on another baking sheet and bake for 6-10 minutes, flipping halfway, until they're slightly crisp.
Spread 1-2 tablespoons of mascarpone cheese over each flatbread. Evenly divide the roasted mushrooms on top and crumble goat cheese over each.
Return the flatbreads to the oven and bake for 6-8 minutes, just until the goat cheese melts slightly and the flatbread edges crisp up.
While the flatbreads bake, whisk the truffle oil and balsamic vinegar with a dash of salt and pepper. Toss with arugula.
Top each flatbread pizza with the dressed arugula and shave fresh Parmigiano Reggiano over the top. If desired, finish with a fried or poached egg on each flatbread. Slice and serve immediately.