Chop kohlrabi leaves. Peel kohlrabi bulbs, and cut in half; thinly slice into half moons.
Heat oil in a large skillet over high heat. Add kohlrabi bulbs, stirring often, about 5 minutes (or until they start to soften). Add kohlrabi leaves, carrots, scallions, and jalapeño to skillet; cook, stirring often, until leaves and bulbs are tender, about 4 to 5 minutes.
Add garlic, lime juice, ginger, brown sugar, and salt to skillet. Cook, stirring constantly about 1 minute. Remove from heat. Sprinkle with cilantro and peanuts