In theory, you could do this on a quarter sheet pan or in a roasting pan in a countertop oven. But for this recipe, you want the added roominess of a standard 16”x13” half-sheet pan so everything can spread in a single layer and truly roast vs. steam.
Arrange rack in the upper third of the oven. Preheat the oven to 425°F.
Place the gnocchi, tomatoes, shallots, and garlic on a sheet pan. Sprinkle with red pepper flakes and salt. Drizzle with olive oil and toss to combine. Spread everything into a single layer.
Bake 30 minutes, stirring halfway through.
Meanwhile, combine the ricotta, goat cheese, salt, and black pepper in a food processor. Process until smooth and well-combined. Add the zest and pulse to combine.
Remove the pan from the oven. Top with arugula. Toss to combine. Bake another 3-4 minutes, or until the arugula has wilted and the tomatoes have burst.
Remove the pan from the oven. Drizzle with balsamic vinegar and stir to combine, scraping up any caramelized bits on the sheet pan.
Serve generously dolloped with the ricotta mixture. Garnish with Parmigiano-Reggiano and basil.