Bring 3 cups water and butter to a boil in a small saucepan. Whisk in grits; cover and cook 5 minutes, stirring frequently. Remove from heat. Stir in cheese, 1/4 teaspoon salt, and 1/2 teaspoon black pepper; cover. Keep warm.
Cook bacon in a large nonstick skillet over medium-high heat until crisp. Add white onion, garlic, and mushrooms to pan; cook 8 minutes or until mushrooms begin to brown and give off liquid, stirring frequently. Add shrimp and red pepper; cook 3 minutes. Combine half-and-half and flour in a small bowl, stirring with a whisk until smooth. Add broth, flour mixture, remaining 3/8 teaspoon salt, and remaining 1/4 teaspoon black pepper to pan; bring to a boil. Cook 2 minutes or until slightly thickened. Top with green onions. Serve shrimp mixture with grits.