For The Pork:
8 garlic cloves, finely chopped
1/4 cup olive oil
3 tablespoons whole grain mustard
3 tablespoons light brown sugar
2 tablespoons kosher salt
2 teaspoons freshly ground black pepper
2 teaspoons oregano leaves, finely chopped
1 1/2 teaspoons smoked paprika
1 skinless, boneless pork shoulder roast, tied (Boston butt; about 6 pounds)
For The Sauce:
1 large shallot, finely chopped
2 cups basil leaves, finely chopped
1 1/4 cups extra virgin olive oil
1/4 cup oregano leaves, finely chopped
1 teaspoon crushed red pepper flakes
1 teaspoon kosher salt
1/2 cup fresh lemon juice
Cook the pork: Mix garlic, oil, mustard, brown sugar, salt, pepper, oregano, and paprika in a small bowl to form a paste. Rub all over pork, then transfer to a resealable plastic bag or slow cooker insert and chill overnight.
Place pork in slow cooker, cover, and cook on low until meat is fork-tender but not yet completely falling apart, 7 1/2–8 hours.
Meanwhile, make the sauce: Mix shallot, basil, oil, oregano, red pepper flakes, and salt in an airtight container. Cover and let rest up to 3 hours at temperature or up to 2 days in refrigerator. Stir in lemon juice just before using.
Transfer pork to a cutting board and let rest 5 minutes. Slice and transfer to a platter. Drizzle with sauce and serve with additional sauce alongside.
Pork can be rubbed 2 days ahead; chill in an airtight container. Sauce, without lemon juice, can be made 2 days ahead; chill in an airtight container.
Both the rub and the sauce can be made in a food processor if desired. To make the pork in the oven, cook at 300°F in a covered Dutch oven, basting occasionally, until an instant-read thermometer inserted into the thickest part of pork registers 145°F (meat will be very tender but not quite falling off the bone), 6–7 hours. If you’re using a larger slow cooker, flip roast at least once and baste occasionally to keep it moist.