Heat olive oil in a large skillet over medium high heat. Once the oil is shimmery, add garlic, ginger, and curry paste. Break the curry apart with a wooden spoon and cook for 2-3 minutes until very fragrant (try not to breathe in the fumes from the cooking curry - it'll clear you out!).
Add coconut milk and chicken broth, scraping any browned bits from the bottom of the skillet. Whisk the curry mixture into the liquid until well incorporated. Now add coconut sugar, kaffir lime leaves (see notes below), and optional fish sauce (see notes below).
Lower the heat to medium (about a 4 out of 10), then add the chicken. Stir to ensure the chicken is covered in the curry sauce. Cover and cook for 5 minutes.
Add the green beans and zucchini, stir, and cover again and cook for another 5 minutes. Remove from heat and stir in basil and lime juice. Allow to rest for 5-10 minutes to finish cooking the veggies. (If you keep cooking them on the heat, they will overcook as they sit in the hot curry sauce.)
Serve over rice, as desired.