In a large skillet, melt the butter over medium heat. Once melted, add the onions, green chilies and pico. Let simmer until the onions are softened, about 5 minutes. Slowly add the flour into the melted butter, stirring constantly to incorporate into the butter. Add the stock slowly, stirring constantly. Once the cooking liquid is smooth, add the half and half, mushrooms, salt and vinegar. Bring to a simmer, stir occasionally.
Once simmering, reduce the heat to medium low and simmer until thickened, about 15 minutes. Stir occasionally to scrape the bottom of the pan. While the sauce is reducing, chop the chicken into bite-sized pieces and heat the oven to 350 degrees.
Pour about 1 cup of chicken stock into a bowl wide enough to dip a tortilla in. Dip the tortilla into the chicken stock and place in large baking dish. Continue this process until the bottom is covered with tortillas, this should take about 6 tortillas. Feel free to tear the tortillas to best fit the shape of your baking dish. Add half of the vegetable mixture on top of the first layer of tortillas, making sure the sauce is evenly distributed. Add half of the chicken and top with a hearty handful or two of cheese.
Add another layer of chicken stock-dipped tortillas. Top with the remaining vegetable mixture and remaining chicken (do not top with cheese). Finish the casserole by adding a final layer of chicken stock dipper tortillas. Cover the tortillas with a final layer of cheese.
Place into the oven and bake at 350 degrees F for 35 to 45 minutes, or until the casserole is warmed all the way through. Remove The King from the oven and let sit for 15 minutes. Cut and serve warm.