Preheat the oven to 425°F (220°C). Line a large rimmed baking sheet with parchment paper or aluminum foil.

Combine the breadcrumbs and 1/2 cup (120 ml) milk in a large mixing bowl. Stir and let the mixture stand for 5 minutes. It should become thick and evenly moistened, without dry crumbs at the bottom.

Add the ground beef, egg, diced onion, garlic, Worcestershire sauce, smoked paprika, 1 teaspoon salt, and 1/2 teaspoon black pepper. Mix gently with clean hands or a fork until the ingredients are evenly distributed. Stop as soon as the mixture holds together; overmixing can make the meatballs firm.

Divide the mixture into 24 portions, approximately 1 1/2 tablespoons each. Roll them gently into balls about 1 1/2 inches (4 cm) wide and arrange them on the prepared baking sheet with a little space between each one.

Place the potatoes in a 5- to 6-quart pot and cover them with cool water by about 1 inch (2.5 cm). Add 1 tablespoon salt. Bring to a boil over high heat, then reduce to a steady simmer.

Bake the meatballs for 18 to 22 minutes. They should be browned around the edges and reach an internal temperature of 160°F (71°C). Avoid baking them longer than necessary because they will cook briefly in the sauce.

Simmer the potatoes for 15 to 18 minutes, or until a knife slides easily through the center of the largest piece without resistance. Do not cook until the potatoes begin falling apart in the water.

While the meatballs and potatoes cook, combine the barbecue sauce, water, apple cider vinegar, and Dijon mustard in a 12-inch skillet. Bring to a gentle simmer over medium-low heat, stirring until smooth.

Transfer the baked meatballs to the skillet with the barbecue sauce. Turn them gently until evenly coated. Simmer over low heat for 5 to 7 minutes, stirring occasionally. The sauce should become glossy and cling to the meatballs rather than pooling thinly around them.

Drain the potatoes thoroughly in a colander: Return them to the warm pot and place it over low heat for about 1 minute, shaking the pot gently. This removes surface moisture that could make the mashed potatoes watery.

Remove the pot from the heat: Add the butter and mash until it has melted into the potatoes. Gradually add the warm milk while continuing to mash. Stop when the potatoes are smooth and creamy with no large lumps.

Fold in 1/2 teaspoon salt and 1/4 teaspoon black pepper. Taste and add more salt only as needed. If the potatoes look too thick, add additional warm milk 1 tablespoon (15 ml) at a time.

Remove the glazed meatballs from the heat and let them stand for 5 minutes. The sauce will thicken slightly and settle around the meatballs.

Divide the mashed potatoes among six plates or shallow bowls. Top each portion with four barbecue meatballs and spoon over additional sauce. Garnish with green onions or parsley, if using.