Preheat the oven to 450 degrees F (235 degrees C). Use a small paring knife to make 6 to 8 slits in the pork. Press a garlic clove into each cut.
Season pork evenly all over with salt and pepper. Place pork in a metal pan or cast iron Dutch oven.
Roast in the preheated oven until well browned, about 30 minutes. Remove pork from the oven and reduce heat to 325 degrees F (165 degrees C)
Add 3/4 cup water and scrape up any bits from the bottom of the pan. Cover with foil.
Return to the oven to bake until fork tender, about 2 1/2 hours. Let pork stand for 20 minutes before shredding.
Meanwhile, for the sauce, combine ketchup, tomato sauce, white vinegar, apple cider vinegar, brown sugar, white sugar, salt, pepper, and hot sauce in a small saucepan and bring to a simmer, stirring often. Cook for 10 to 15 minutes. Adjust seasoning to taste.
Serve sauce with shredded pork.