Make the green curry paste: combine the green chilies, diced shallot, minced garlic, minced ginger, chopped lemongrass, ground coriander, white pepper, soy sauce, fish sauce, sugar, and 3 tablespoons coconut milk in a food processor and process until fragrant and smooth. Set aside.
Heat the wok or large skillet over high heat, then add the vegetable oil and swirl to coat.
Add the prepared green curry paste to the hot oil and stir-fry briefly for about 1 minute until fragrant.
Pour in about 3/4 of the reserved coconut milk (about 1/2 cup), stirring to combine with the paste and form the curry sauce.
Add the chopped chicken thighs, stir to coat with the curry sauce, and bring to a boil; then reduce heat to a simmer.
Simmer the chicken for about 8 minutes, stirring occasionally, until cooked through.
Stir in the zucchini and thin strips of kaffir lime leaves, then simmer for another 3 minutes until the zucchini is tender-crisp.
Divide the curry into bowls, sprinkle with chopped Thai basil, and serve with lime wedges. Drizzle about 2–3 tablespoons of the reserved coconut milk over each bowl if desired.
To cook the rice: rinse the jasmine rice, then combine it with 2 cups water, 1 tablespoon butter, and 1 teaspoon salt in a pot and bring to a boil.
Reduce heat to low, cover, and simmer the rice for 18–20 minutes. Remove from heat, fluff with a fork, and serve alongside the curry.