Cook the gnocchi according to package directions, reducing the stated cook time by about 1 minute so they are just tender (about 2 minutes total for many mini gnocchi). Drain well in a colander and toss with 1 tablespoon olive oil to prevent sticking.
Heat a large nonstick skillet over medium heat and add 2 tablespoons olive oil. When the oil is hot, add the gnocchi in a single layer (work in batches if needed) and cook without stirring until bottoms are golden and crisp, then flip and crisp the other side. Remove with a slotted spoon to a paper-towel-lined plate.
In a small saucepan or small skillet over medium heat, melt the butter and begin whisking or stirring. Continue cooking until the butter foams and small brown bits appear and the aroma is nutty, about 2–3 minutes; remove from heat immediately.
Stir the minced garlic and all the chopped fresh herbs (basil, chives, oregano, rosemary, thyme) into the warm brown butter and mix to combine.
Return the toasted gnocchi to the skillet or place in serving bowls, drizzle with the herb brown butter, sprinkle with finely grated Parmesan and additional fresh herbs as desired, and serve immediately.