Heat oil in a medium saucepan over medium heat and stir in the garlic, lemongrass and ginger until fragrant, about 30 seconds. Keep stirring often and make sure not to burn the garlic. Add in the curry paste and ¼ cup of the coconut milk. Whisk together until the curry paste is dissolved. Add in the rest of the coconut milk, brown sugar, soy sauce, lime juice and chili paste. Reduce to a simmer and let cook until slightly thickened, about 10 minutes.
While the sauce is cooking, add in the linguine and cook until just al dente. Right before the linguine is done (about 2 minutes before estimated cooking time), add in the broccoli, carrots, edamame and red bell pepper. Drain, toss with sauce and serve.