Step 1Heat the olive oil in a 3-quart saucepan over medium heat. Add the garlic, ginger and red pepper flakes and cook until fragrant, 1 minute. Add the coconut milk and turmeric, bring to a boil, then reduce the heat to low and simmer until the mixture thickens slightly and has the consistency of half-and-half, about 30 minutes. Stir in the brown sugar and season to taste with salt and pepper.
Step 2Meanwhile, fill a large saucepan with water and bring to a boil. Add 1 tablespoon of salt. Add the carrots and blanch 2 minutes. Remove to a bowl of ice water. Add the bell peppers to the boiling water and blanch 2 minutes, then remove to the bowl of ice water; do the same for the broccoli. Drain the vegetables and set aside.
Step 3Bring another saucepan full of water to boil and add 1 tablespoon of salt. Add the pasta and cook until al dente, about 2 minutes. Drain and keep warm.
Step 4Heat the vegetable oil in a wok or large skillet over medium-high heat. Add the shrimp and cook, stirring, until it turns opaque, 3 to 5 minutes. Add the reserved vegetables and cook until heated through. Add the finished sauce and cook until slightly reduced, about 1 minute. Add the pasta and toss to coat. Serve garnished with the cilantro or green onions.