In a large skillet or wok, heat 1 tablespoon vegetable oil over medium-high. In 2 batches, saute brussels sprouts until tender, 5 minutes total; season with salt and pepper. Transfer to a platter. Add bean sprouts to pan and cook until heated through, 1 minute; add to platter.
Wipe out pan, add remaining tablespoon vegetable oil, and return to heat. Add garlic, chile, and scallion whites and cook until fragrant, 30 seconds. Add shrimp, season with salt and pepper, and saute until opaque throughout, about 4 minutes. Add to platter, drizzle with sesame oil, top with scallion greens, and toss. Serve with lime wedges.