Heat oil in a wok until it starts to shimmer. Add curry paste, garlic, and ginger; stir together and fry until fragrant, 2 to 3 minutes. Add chicken breast slices, stirring until coated. Cook until just slightly pink in the center, 3 to 5 minutes.

While chicken is cooking, place baby corn and bamboo shots in a small saucepan, cover with water, and place over high heat. Bring to a boil. Drain then cover with water again.

Pour coconut milk into the wok. Bring to a slow simmer, stirring occasionally, about 5 minutes. Drain corn and bamboo shoots and add in. Stir in fish sauce, palm sugar, and kaffir lime leaves. Bring back up to a simmer and cook for 5 minutes. Stir in bell pepper strips and cook until crisp-tender, about 3 minutes.

Remove curry from heat and stir in Thai basil.