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 or large skillet and grease with the 1 Tbsp. of oil. Season the shrimp with salt and pepper and brush both sides using the 1/3 cup of barbecue sauce. Cook the shrimp over moderate heat, turning once, until cooked through, about 4 minutes total.
To Make the 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, approximately 20 minutes.
Remove the saucepan from the heat and stir in the cheese, butter and cream. Season with salt and pepper.
To Serve
Mound the cheese grits on a plate and top with shrimp and garnish with some parsley. Pass the barbecue sauce.
Notes
Make Ahead: The barbecue sauce can be made ahead and refrigerated for up to a week.I like the Polenta Corn Grits but you can also use the White Southern style.