Heat the olive oil in a large deep skillet or Dutch oven over medium-high heat.
Add the chicken pieces and cook for 4-5 minutes until lightly browned but not fully cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 30 seconds until fragrant.
Stir in the rinsed rice and cook for 1-2 minutes, stirring frequently, to lightly toast the rice.
Add the chicken broth, milk, diced green chilies, ground cumin, smoked paprika, chili powder, salt, and black pepper. Stir to combine.
Return the chicken pieces to the skillet, spreading them evenly over the rice mixture.
Bring the mixture to a boil, then reduce heat to low. Cover and simmer for 18-20 minutes, or until the rice is tender and the liquid is absorbed.
Remove the skillet from heat and sprinkle the shredded cheddar and Monterey Jack cheeses evenly over the top.
Cover the skillet again and let it sit for 5 minutes to allow the cheese to melt.
Gently fluff the rice and chicken with a fork to combine the melted cheese throughout.
Garnish with chopped fresh cilantro and sliced jalapeño if desired before serving.