How to Make It
In a large pot of salted boiling water, cook the lasagnette until al dente. Drain the pasta, reserving 1 cup of the cooking water. Transfer the pasta to a baking sheet and toss with 2 teaspoons of the olive oil.
In a medium saucepan of salted boiling water, blanch the peas until crisp-tender, about 3 minutes. Using a slotted spoon, transfer the peas to a plate to cool. Add the asparagus to the saucepan and blanch until they are crisp-tender, about 2 minutes. Drain the asparagus and add them to the peas.
Preheat the broiler and butter a shallow 9-by-13-inch baking dish. In a large, deep skillet, heat the remaining 3 tablespoons of olive oil until shimmering. Add the mushrooms, season with salt and pepper and cook over high heat, stirring occasionally, until browned, about 5 minutes. Add the shallots, thyme and the 2 tablespoons of butter and cook, stirring, until the butter is melted and the shallots are softened, about 3 minutes. Stir in the wine and lemon juice. Remove from the heat; stir in the goat cheese until melted.
Fold the asparagus, peas, mint and 1/4 cup of the Parmigiano into the sauce and season with salt and pepper. Add the pasta and the reserved cooking water and toss until thoroughly coated. Transfer the pasta to the prepared baking dish and sprinkle the remaining 1/4 cup of Parmigiano on top. Broil the lasagnette 6 inches from the heat until lightly golden and bubbling, 3 to 5 minutes. Let rest for 5 minutes before serving.