Heat 1 tablespoon coconut oil in a deep 12-inch skillet over medium-high heat until shimmering.
Add the chicken pieces and cook about 5 minutes, stirring occasionally, until the outside is no longer pink; transfer chicken to a plate and set aside.
In the same skillet, add the remaining 1 tablespoon coconut oil, then stir in the green curry paste, grated garlic, grated ginger, and grated lemongrass until fragrant.
Add the sliced onion and red bell pepper and cook about 5 minutes, stirring, until they begin to soften.
Pour in the chicken stock, bring to a boil, then reduce heat and simmer about 3 minutes.
Stir in the coconut milk, snow peas, and the reserved chicken; keep at a gentle simmer and cook about 5 minutes, until heated through and snow peas are just tender—do not boil.
Remove from heat and stir in the sliced basil, lime zest, and lime juice.
Serve the curry hot over cooked rice.