Place the milk and water in a saucepan with salt and pepper, and bring to a simmer over high heat. Reduce the heat to medium and slowly add the grits, stirring constantly. Reduce the heat to medium-low and simmer, stirring frequently for about 8 minutes until the grits are smooth and creamy. Stir in the butter, heavy cream, and Parmesan until the butter is melted and the grits are hot and very creamy.
While the grits are simmering, season the chicken breasts with salt and pepper. In a large skillet, heat the olive oil over medium high heat. Add the chicken breasts and sauté until done, about 4 to 5 minutes on each Remove the chicken breasts to a plate and set aside. Do not wipe out the skillet!
Add the garlic to the skillet and sauté over medium heat for 1 minute until it starts to color. Add the green chilis, turn the heat to medium high, then add the broth and stir to scrape up all the little flavorful bits that may be stuck to the bottom of the skillet. Simmer for about 4 minutes until the mixture reduces slightly. Whisk in the sour cream until well combined and heat just until the mixture is hot, but don’t allow it to come to a simmer.
Scoop some grits onto each of 4 individual plates. Place a chicken breast on each, on top of the grits, and pour the sauce over the chicken breasts. Serve hot.