Prepare noodles according to package directions; drain and set aside.
Heat the oil in a non-stick pan over medium-high heat. Add the chicken, garlic, and ginger. Sauté until the chicken starts to lightly brown.
Add the broth, milk, peanut butter, soy sauce, and lime juice. Turn down the heat and bring to a simmer. Add in the broccoli and carrots, cover with a lid and simmer until tender. Toss in the noodles. (Taste and adjust the seasoning with more lime juice or salt.)