Prep all ingredients: cut chicken into 1.5-inch pieces, cut eggplant into wedges and place in water, julienne bell pepper, trim snow peas, mince garlic, grate ginger, bruise lemongrass, tear lime leaves, and measure out fish sauce, sugar, and curry paste.
Heat wok over medium heat and add 1 tablespoon coconut oil. Fry green curry paste for 2-3 minutes, stirring constantly, until fragrant and slightly darkened. Add garlic and ginger and stir for 30 seconds.
Add chicken pieces to the wok and cook for 2-3 minutes without stirring to get a light sear. Stir and toss to coat with paste, cooking another 1-2 minutes until chicken is mostly opaque on the outside.
Add bruised lemongrass and chicken broth to the wok. Bring to a gentle simmer and cook for about 3 minutes to infuse the lemongrass flavor.
Pour in the full can of coconut milk and add the torn kaffir lime leaves. Bring back to a gentle simmer and cook for 5 minutes, stirring occasionally, until the sauce turns golden-green.
Add drained eggplant and julienned red bell pepper. Continue simmering over medium-low heat for 8-10 minutes until eggplant is tender and creamy and chicken is fully cooked.
Stir in fish sauce and palm sugar. Taste and adjust seasoning, adding more fish sauce, sugar, or a squeeze of lime as needed. Add sliced chili for extra heat.
Add snow peas (or spinach) and cook for 1-2 minutes until just tender. Turn off heat and stir through fresh Thai basil leaves. Remove and discard lemongrass pieces before serving.
Spoon curry generously over steamed jasmine rice. Garnish with additional Thai basil, sliced red chili, and lime wedges. Serve immediately while hot.