To make the green curry paste, toast coriander seeds, cumin seeds, and white peppercorns in a dry skillet until fragrant, about 1-2 minutes. Transfer to a mortar and pestle or spice grinder and grind into a fine powder.
Add the green chilies, garlic, shallot, lemongrass, galangal or ginger, cilantro roots or stems, and shrimp paste (if using) to the mortar and pestle or food processor. Grind or blend into a smooth paste, adding a little water as needed to make it easier to work with.
Heat vegetable oil in a large pot or wok over medium-high heat. Add the green curry paste and stir-fry until fragrant, about 1-2 minutes.
Add the chicken and stir-fry until browned, about 3-5 minutes.
Pour in the coconut milk and chicken stock or water. Stir well to combine and bring to a simmer.
Add the fish sauce and palm sugar or brown sugar, and stir to dissolve.
Add the Thai eggplant, bamboo shoots, green beans, and torn kaffir lime leaves. Stir to combine and simmer until the vegetables are tender and the chicken is cooked through, about 10-15 minutes.
Add the torn basil leaves and stir to combine.
Taste and adjust the seasoning as needed, adding more fish sauce for saltiness, sugar for sweetness, or curry paste for heat.
Serve hot with steamed rice and enjoy your delicious Thai Green Curry!