Bring a 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 take roughly 2 minutes total for these mini gnocchi). Drain well in a colander and return to the pot; drizzle 1 tablespoon olive oil over the gnocchi and toss to prevent sticking.
Heat a large nonstick skillet over medium heat and add 2 tablespoons olive oil. When hot, add the gnocchi in a single layer (work in batches if needed) and cook without moving until the bottoms are golden and crisp, then flip and brown the other side. Remove with a slotted spoon to a paper towel-lined plate to drain excess oil.
While the gnocchi cooks, melt the butter in a small saucepan or skillet over medium heat. Whisk or stir continuously; after 2–3 minutes the butter will foam and brown bits will form on the bottom. Remove from heat as soon as it turns golden-brown and smells nutty, and continue whisking for about 30 seconds.
Stir the chopped basil, chives, oregano, rosemary, thyme, and minced garlic into the browned butter until combined.
Divide the toasted gnocchi among plates or bowls, drizzle with the herb brown butter, sprinkle with finely grated Parmesan, and add a few extra fresh herb leaves if desired. Serve immediately.