Heat the vegetable oil in a large cast-iron skillet or other heavy skillet over medium-high heat. Season the chicken thighs with salt and pepper and add to the skillet; cook until browned, about 4 minutes per side. Remove to a plate.
Remove and discard 3 tablespoons rendered fat from the skillet; reduce the heat to medium. Add the garlic, shallot and ginger and cook until softened, 2 to 3 minutes. Add the coconut water, fish sauce and brown sugar and stir until the sugar is melted. Bring to a simmer and cook until the sauce thickens (the bubbles will begin to slow as it thickens), 4 to 5 minutes.
Return the chicken and any collected juices to the skillet and stir to coat the chicken. Cook, reducing the heat if the sauce starts thickening too much, until the chicken is cooked through, 5 to 6 minutes. Remove from the heat and stir in the jalapeños. Serve the chicken over the rice and top with cilantro leaves.