Bring a large pot of salted water to a boil and cook the gnocchi according to package directions, reducing the stated time by about 1 minute (they should be just tender).
Drain the gnocchi and drizzle with 1 tablespoon olive oil; toss gently to prevent sticking.
Heat a large nonstick skillet over medium heat and add 2 tablespoons olive oil. When hot, add gnocchi in a single layer (work in batches if needed) and cook until golden and crisp on both sides, turning as necessary. Remove with a slotted spoon and drain briefly on paper towels.
Meanwhile, melt the unsalted butter in a small saucepan or skillet over medium heat and whisk immediately. Continue cooking and whisking until the butter foams and brown bits appear on the bottom, about 2–3 minutes, then remove from heat.
Stir the minced garlic and all the chopped fresh herbs (basil, chives, oregano, rosemary, thyme) into the warm brown butter and whisk briefly to combine.
Divide the toasted gnocchi among plates or bowls, drizzle generously with the herb brown butter, sprinkle with finely grated Parmesan and additional fresh herbs if desired, then serve immediately.
