Preheat a smoker to 250°F. In a small bowl, stir together the barbecue sauce and brown sugar until the sugar is mostly dissolved. Set the glaze aside.

In a large bowl, whisk the egg with the milk. Add the breadcrumbs, diced onion, garlic powder, smoked paprika, salt, and black pepper. Stir to combine, then add the ground beef and mix gently until evenly combined.

Place the meat mixture on a sheet of parchment and shape it into a rectangle about 1 inch thick. Lay the cheddar stick lengthwise down the center, leaving a border at each end. Fold the meat around the cheese and pinch the seams and ends closed. Shape into a compact loaf with the seam underneath.

Set the meatloaf on a smoker-safe tray or a sheet of foil with the edges turned up. Smoke until the center reaches 150°F, then brush the top and sides with about half of the brown sugar barbecue glaze.

Continue smoking until the center reaches 160°F. Brush with the remaining glaze during the last 10 minutes of cooking. Let the meatloaf rest for 10 minutes before slicing so the cheese is less likely to spill out.