Generously salt the goose leg quarters all over, then sear them in a large skillet with 2 tablespoons of the olive oil over medium-high heat for 2 minutes on each side until nicely browned. Transfer the browned goose leg quarters to the slow cooker.
In the same skillet over medium heat, add the remaining one tablespoon of olive oil with the onions and sauté for about 4 minutes, or until starting to soften. Then add the minced garlic and continue to sauté for another 1 minute.
Now separate the pineapple juice from the chunks. Open the can of pineapple chunks and pour out the juice into a small bowl. Place the pineapple chunks in the refrigerator until ready to use later in the recipe.
Transfer the onions and garlic to the slow cooker with the beef broth, pineapple juice, black pepper, and smoked paprika. Cover and cook on Low for 8 to 10 hours or on High for 5 to 6 hours.
Shred the cooked goose leg quarters with tongs or a fork. Discard the bones.
Now make the caramelized pineapple. In a skillet over medium heat, combine the pineapple chunks and teriyaki sauce. Cook, stirring constantly, for about 5 minutes.
Spoon about ⅓ cup of the pulled goose meat onto each brioche bun, then top with one tablespoon of BBQ sauce and some caramelized pineapple chunks. You can serve as is, or garnish with some microgreens or fresh spinach.