Bring 1 1/2 cups water to a simmer in a small saucepan. Stir in the rice and a pinch each of salt and pepper. Reduce the heat to low, cover and cook until the water is absorbed and the rice is tender, about 15 minutes; set aside.
Meanwhile, heat the olive oil in a large pot over medium-high heat. Add all but 1/4 cup of the chopped onion, the jalapenos and garlic; season with salt and pepper. Cook, stirring, until the onion is tender, about 5 minutes. Add the turkey, chili powder and cumin. Cook, breaking up the meat, until browned, about 5 minutes. Stir in the tomatoes; season with salt. Continue cooking, stirring occasionally, until the tomatoes start breaking down, about 5 minutes. Add the hominy and 1 1/2 cups water. Bring to a simmer and cook until thickened, about 10 minutes.
Stir the jalapeno brine and half of the cilantro into the chili; season with salt and pepper. Fluff the rice with a fork. Serve the chili over the rice; top with the reserved chopped onion and cilantro and more pickled jalapenos.