Start by cutting chicken into bite-size pieces. Heat skillet over medium-high heat and add 1 Tablespoon of oil. Cook chicken for about 4 minutes, stirring often, until no longer pink. This chicken will continue to cook in the rice so it doesn't need to be fully cooked. Generously season the chicken with salt and pepper. Remove the chicken from the skillet and cover.
Add remaining 1 Tablespoon of oil to the skillet. Saute the onion for about 5-6 minutes or until it begins to soften. Add garlic and cook for 1 minute longer. Sprinkle with thyme and tarragon.
Add in jasmine rice, chicken broth, milk, and chicken. Let it come to a simmer. Reduce the heat to LOW to allow the rice to cook low and slow. Cover with a lid and cook for at least 20-25 minutes or until the rice is fully cooked and softened. The rice should have absorbed a lot of the liquid but it will still be creamy.
Stir in parmesan and cheddar cheeses. Taste for seasonings and add more salt and pepper as needed.