Bring a large sauce pot of water to a boil. Cook gnocchi according to package directions. Reserve ½ cup pasta water. Drain. Transfer gnocchi to a bowl and drizzle with olive oil. Set aside.
Heat 2 teaspoons of olive oil in a large skillet on medium-high heat. Add cooked gnocchi. Cook 5 to 7 minutes, or until lightly browned and crispy on both sides. Make sure to toss gnocchi regularly to prevent burning. Remove from pan.
In the same pan, heat 1 tablespoon of olive oil. Add the onion, garlic, ginger, and bok choy stems. Cook 2 to 3 minutes, or until vegetables begin to soften.
Add the butter and let brown. Add the miso paste, reserved pasta water, and pepper. Bring to a boil and reduce to a simmer, stirring frequently. Cook 3 to 4 minutes, or until sauce thickens and is reduced by half.
Add the bok choy leaves. Cook 1 minute, or until just begin to wilt. Stir in gnocchi. Toss to coat. Remove from heat and sprinkle with green onions.