In a large skillet, heat the vegetable oil over medium-high heat. Add the chicken pieces and season with salt and pepper. Cook until browned and cooked through, about 5-7 minutes.
Add the sweet chili sauce and soy sauce to the skillet, stirring to coat the chicken evenly. Cook for an additional 2-3 minutes until the sauce thickens slightly.
In a separate pot, steam the red bell pepper and snap peas until tender, about 3-4 minutes.
In a small bowl, whisk together the coconut milk, lime juice, and lime zest.
To assemble the bowls, divide the cooked jasmine rice among serving bowls. Top with the sweet chili chicken, steamed vegetables, and drizzle with the coconut lime sauce.
Garnish with chopped green onions and cilantro before serving.