Slice the scallions thinly, separating the green and white parts as you go. Heat the oil in a large skillet over high heat for 30 seconds. Add the whites of the scallions and sauté for 30 seconds. Add the mushrooms, season with salt and pepper, and sauté, stirring occasionally, until they soften completely and brown in places, about 4 minutes. Stir in 1 tsp. lemon juice, the mint, and the scallion greens. Taste and add salt, pepper, and lemon juice as needed.
To assemble, spoon the mushrooms onto the crostini, top with a Parmigiano curl, and serve.