1. Heat the butter and oil in a pan and add the jalapenos, onions and bell peppers. Cook until softened, 5 minutes or so.
2. Pour in the grits, followed by the broth and 3 cups water. Stir, and then bring to a boil. Reduce the heat to low, and then cover and cook for 30 to 40 minutes, stirring occasionally.
3. Add the half-and-half milk. Stir, and then cook until the grits are tender, another 20 to 30 minutes. At any time, add more broth if the grits look like they need more cooking liquid.
4. Remove the grits from the heat and stir in the cheese. Serve immediately.