For the BBQ beef filling:

1. In a skillet, cook the ground beef over medium heat until browned, breaking it up into small crumbles as it cooks. Drain excess fat if necessary.
2. Add the chopped onion and minced garlic to the beef and cook until the onions are soft and fragrant, about 3-4 minutes.
3. Stir in the BBQ sauce and cook for another 2 minutes, allowing the beef to absorb the sauce. Season with salt and pepper to taste. Remove from heat and set aside to cool slightly.
4. Stir in the shredded cheese and mix well.

For the mashed potato cups:

1. In a large bowl, combine the mashed potatoes, egg, and flour. Season with salt and pepper. Mix until smooth and well combined.
2. Roll the mashed potato mixture into 12-15 golf-ball-sized portions.
3. Flatten each ball into a small patty using your hands or a spoon, creating a well in the center to hold the BBQ beef filling.
4. Spoon about 1-2 tablespoons of the BBQ beef mixture into the well of each mashed potato cup.
5. Carefully fold the edges of the mashed potato around the beef, sealing it into a cup shape.
6. Roll each filled potato cup in breadcrumbs to coat them lightly.

To fry:

1. Heat vegetable oil in a deep fryer or large pot to 350°F (175°C). The oil should be deep enough to fully submerge the cups.
2. Gently lower the mashed potato cups into the hot oil, a few at a time, and fry for 3-4 minutes, or until golden brown and crispy on the outside.
3. Remove the cups from the oil using a slotted spoon and place them on paper towels to drain any excess oil.

To serve:

1. Serve the deep-fried mashed potato cups warm, with extra BBQ sauce for dipping if desired.