Heat the olive oil in a 12-inch (30 cm) cast-iron or nonstick skillet over medium-high heat until it shimmers, about 2 minutes.
Add the gnocchi in a single layer, working in two batches if the pieces don't fit without touching. Season with 1/4 tsp salt and the black pepper.
Cook undisturbed for 3 to 4 minutes until the bottoms turn deep golden and blistered. Resist the urge to stir, that's what builds the crust.
Flip each piece and cook 3 to 4 minutes more until crisp on two sides and tender when you bite into one. Transfer the gnocchi to a plate.
Wipe out the skillet, reduce the heat to medium, and add the butter. Swirl as it melts, then keep cooking 2 to 3 minutes, swirling often, until it turns amber and smells nutty with browned specks on the bottom of the pan.
Add the garlic and sage leaves right away. They'll crisp and darken slightly in 30 to 45 seconds, watch closely so the sage doesn't turn bitter.
Return the crisped gnocchi to the skillet and toss for about 1 minute to coat everything in the brown butter, sage, and garlic.
Pull the skillet off the heat. Add the parmesan, a pinch of red pepper flakes if using, and a squeeze of lemon, then toss once more.
Plate immediately with extra parmesan and sage on top. Serve hot, the crust softens as it sits.