2 tablespoons active dry yeast
1/2 cup plus 1 tablespoon sugar, divided
1/2 cup warm water, 105°F to 115°F
1 cup whole milk
1 cup (2 sticks) unsalted butter, divided
2 large eggs, room temperature and lightly beaten
2 teaspoons salt
5 cups all-purpose flour, plus more if needed
24 smoked cocktail sausages
Additional flour, for rolling
Sprinkle yeast and 1 tablespoon sugar over the warm water and let sit until foamy, about 10 minutes.
Meanwhile, heat the milk and 1/2 cup (1 stick) butter in a small saucepan over medium-low until butter is just melted. Remove from heat and stir in the remaining 1/2 cup sugar until dissolved.
Combine the milk mixture, yeast mixture, eggs, and salt in a large mixing bowl. Stir in 4 cups of flour and mix well. Add the remaining flour 1/2 cup at a time, until a ball of dough forms — the dough should be smooth and fairly sticky, but not tacky; if it seems too wet, add another 1/2 cup of flour.
Transfer the ball to a large greased mixing bowl. Cover with plastic wrap or a damp tea towel and set it in a warm, draft-free place. Let the dough rise until doubled in size, about 1 1/2 hours.
Preheat oven to 350°F. Place cocktail sausages on a baking sheet and cook for 20 minutes. Remove from the oven and allow to cool. Thoroughly blot with a paper towel to remove any excess grease.
Meanwhile, grease two 9-inch cake pans or disposable aluminum rounds. Turn out the dough onto a lightly floured surface. Sprinkle a little bit of flour on top of the dough and roll to 1/2-inch thickness. Using a floured biscuit cutter, cut dough into 2-inch rounds.
Gently press each round into an oval shape, place 1 cocktail sausage in the center, and fold the dough around to form a half moon. Gently press the very center front to seal shut. Set aside and continue with remaining rounds.
Melt the remaining 1/2 cup (1 stick) butter. Using a silicone pastry brush, lightly coat each roll with butter. Starting on the outside and working your way in, place the rolls, seam-side out, into the greased pans (about 12 rolls per pan, see above image). Cover the pans with plastic wrap or a damp tea towel and let rise in a warm, dry place until doubled in size, 30 minutes to 1 hour.
Preheat oven to 375°F. Bake the rolls, uncovered until just barely golden, 15 to 20 minutes. Remove rolls from the oven and brush with any remaining melted butter. Serve hot.