Open the coconut milk without shaking; spoon off 0.5 cup of the thick cream and set both the reserved cream and the remaining milk aside.
Heat a wok or wide saucepan over medium heat. Add the neutral oil and the reserved cream, then stir in the red curry paste. Cook, stirring, until fragrant and the fat begins to split, 2–4 minutes.
Stir in the natural creamy peanut butter and cook 1 minute to combine.
Add the boneless skinless chicken thighs, cut into bite-size pieces, and cook, stirring, until opaque on the outside, 2–3 minutes.
Pour in the remaining milk and the water; bring to a gentle simmer.
Season with the fish sauce and palm sugar; add the makrut lime leaves. Simmer, stirring occasionally, until the chicken is cooked through and the sauce lightly thickens to coat a spoon, 8–12 minutes.
Stir in the red bell pepper and the Thai bird's eye chile (if using) and simmer until the vegetables are crisp-tender, 2–3 minutes.
Turn off the heat; stir in the fresh lime juice and half of the Thai basil leaves, reserving the rest for garnish. Taste and adjust seasoning to balance salty, sweet, and tart.
Serve over cooked jasmine rice and garnish with the reserved leaves and the roasted peanuts.