Barbecue Shrimp with Cheese Grits

2 tablespoons unsalted butter
1 small onion, cut into 1/4-inch dice
2 garlic cloves, very finely chopped
1 1/4 cups ketchup
1/4 cup bourbon
3 tablespoons cider vinegar
2 tablespoons molasses
2 tablespoons honey
2 teaspoons Tabasco
1 teaspoon chopped thyme
1/2 teaspoon cayenne pepper
Kosher salt and freshly ground black pepper
1 1/2 pounds peeled and deveined large shrimp
1 tablespoon vegetable oil
Cheese Grits, for serving

Cheese Grits:
4 cups low-sodium chicken broth
1 garlic clove, minced
1 cup old-fashioned grits
4 ounces extra-sharp cheddar cheese, shredded (1 1/2 cups)
4 tablespoons unsalted butter
2 tablespoons heavy cream
Kosher salt and freshly ground pepper

In a medium saucepan, melt the butter. Add the onion and cook over moderate heat until softened, about 5 minutes. Add the garlic and cook until fragrant, about 1 minute. Stir in the ketchup, bourbon, cider vinegar, molasses, honey, Tabasco, thyme and cayenne. Simmer over low heat, until thickened, about 40 minutes.
Transfer the barbecue sauce to a blender and puree until smooth. Season with salt and pepper. Pour 1/3 cup of the sauce into a small bowl and reserve the rest.

Preheat a grill pan. Season the shrimp with salt and pepper and brush on both sides with the 1/3 cup of barbecue sauce. Grease the grill pan with the oil. Grill the shrimp over moderate heat, turning once, until cooked through, about 4 minutes. Transfer to plates and serve with cheese grits and the remaining barbecue sauce.
MAKE AHEAD The barbecue sauce can be refrigerated for up to a week.

For cheese grits:
In a medium saucepan, bring the chicken broth to a boil. Add the garlic and slowly stir in the grits. Reduce the heat to moderately low and cook, stirring frequently, until the grits are tender, 20 minutes. Remove the saucepan from the heat and stir in the cheese, butter and cream. Season with salt and pepper and serve immediately.

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