Separate the coconut milk. Without shaking the can, open it and scoop out the thick cream from the top into a small bowl. Reserve the remaining thin coconut milk separately.
Stir-fry the vegetables. Heat the oil in a large skillet or wok over medium-high heat. Add the eggplant and stir-fry for 3-4 minutes until starting to soften and pick up some color. Add the bamboo shoots and cook another minute. Transfer to a plate and set aside — this step builds deeper flavor in the vegetables.
Fry the curry paste. In the same skillet, add the coconut cream (the thick part) and cook for 2-3 minutes until it begins to separate and you see oil pooling on the surface. Add the green curry paste and fry for 1-2 minutes, stirring constantly, until very fragrant.
Cook the chicken. Add the sliced chicken to the paste and stir to coat. Cook for 3-4 minutes until mostly white on the outside.
Build the sauce. Pour in the reserved thin coconut milk and chicken broth. Stir in the makrut lime leaf powder. Bring to a gentle boil, then reduce to a simmer.
Combine and simmer. Return the stir-fried vegetables to the pan. Simmer for 6-8 minutes until the chicken is cooked through and the sauce has thickened slightly. Add the red bell pepper in the last 2-3 minutes (you want it slightly crisp).
Season. Stir in the fish sauce and palm sugar. Taste and adjust for a balance of salty, sweet, and spicy.
Serve. Ladle the curry into bowls over steamed jasmine rice. Top with fresh Thai basil leaves, sliced red chili, and a squeeze of lime.