Sear the gnocchi. In a large skillet set over medium heat, heat 1 tablespoon of olive oil for about 30 seconds or until shimmering. Working with one package at a time, add the gnocchi in a single layer and cook, undisturbed, until golden brown on one side, about 2 minutes. Flip and continue to cook until evenly golden brown, about 2 minutes more. Transfer to a plate and set aside. Repeat with the remaining package of gnocchi, adding a second tablespoon of olive oil to the pan.
Sauté the vegetables. Add the remaining 1 tablespoon olive oil and heat until shimmering. Add the cherry tomatoes, roasted red peppers, onion, garlic, lemon zest, salt, black pepper, and red pepper flakes. Cook for 7 to 9 minutes, stirring occasionally, until the tomatoes begin to burst and onions have softened.
Combine and serve. Reduce the heat to low. Add the crispy gnocchi, basil, and Parmigiano-Reggiano and toss to combine. Taste and season with more salt and pepper, if necessary. Serve immediately, topped with more basil and Parmigiano-Reggiano.
