Heat oil in large skillet over medium heat, add garlic, toss well. Add curry paste and cook, mashing and stirring to dissolve it, until fragrant and softened, about 2-3 minutes.
Crumble in ground beef and then add mushrooms. Cook, tossing often, until meat and mushrooms are browned and seasoned with curry paste, 2-3 minute.
Add chicken broth, green beans, zucchini, lime leaves (if using), fish sauce, sugar, and salt and bring to gentle boil. Cook 3-5 minutes, until veggies are tender and meat is cooked through. Stir in basil leaves and remove from heat. Transfer to serving dish & serve hot or warm.