Reserve 2 tablespoons of the cilantro leaves and place the rest of the cilantro, jalapeños, ginger in a food processor and pulse until finely chopped. Heat 2 tablespoons of oil in a large saucepan over medium heat and cook the ginger mixture for 2 minutes or until fragrant and season with 1/2 teaspoon kosher salt. Add the onion and red bell pepper and cook until softened, about 5 minutes.
Add the coconut milk and soy sauce and bring to a simmer. Add the chicken and the remaining salt and simmer until cooked through, about 10 minutes, and so the sauce thickens.
Spoon rice into bowls and serve with the ginger chicken. Garnish with the remaining cilantro leaves, green onion, and sliced serrano peppers.