Add the beef to a medium bowl and sprinkle with the baking soda and 1/2 teaspoon salt. Using your hands or a wooden spoon, work the mixture until well combined.
Let the beef mixture rest on the counter for 20 minutes to let the baking soda works its tenderizing magic.
In a large skillet over medium-high heat, heat the oil until shimmering. Add the onion, 1/2 teaspoon salt, and the pepper. Cook, stirring often, until the onions are softened but not browned, 4 to 6 minutes.
Add the garlic and paprika and cook for 1 minute more.
Add the beef to the skillet and cook, breaking up the meat with a wooden spoon, until it’s mostly brown, 5 to 10 minutes.
Add 1/4 cup of the bourbon, the tomato sauce, tomato paste, Worcestershire sauce, brown sugar, and mustard. Bring to a boil and then reduce the heat to low and simmer, uncovered, until the sauce has thickened, 15 to 25 minutes.
Taste and adjust the seasoning accordingly with salt, pepper, and up to a tablespoon or so more bourbon, if desired.
Spoon the sloppy joes onto the buns and top each with shredded Cheddar and pickled onions.
Serve hot. Don’t forget napkins aplenty.