1. Make dough: Stir together yeast and warm water in a small bowl; let stand until foamy, about 5 minutes.

2. In the bowl of a standing mixer fitted with the whisk attachment, mix milk, butter, eggs, sugar, salt, and vanilla on low speed until eggs and milk are combined, 2 minutes. Add yeast mixture; mix 1 minute. Add 4 cups flour; mix until smooth.

3. Switch to the dough hook; continue mixing, and add more flour, 1/2 cup at a time, just until dough no longer sticks to bowl (about 2 more cups). Beat dough on medium-high speed, 10 minutes.

4. On a lightly floured work surface, knead the dough until very smooth, about 2 minutes; transfer to a large buttered bowl. Cover dough with a clean kitchen towel; let rise until doubled in bulk, about 1 hour.

5. Make topping: Melt butter in a saucepan; stir in sugars and corn syrup. Cook, stirring, over low heat, until sugar dissolves, about 5 minutes. Set aside.

6. Punch down dough, then turn out onto a lightly floured work surface. Divide dough into three equal pieces; roll each into a ball. (Work with one ball at a time; refrigerate or freeze any not being used.)

7. Assemble buns: Roll out dough ball into a 10-by-13-inch rectangle. Brush dough with melted butter. Top with filling; leave a 1/2-inch border. Starting on one long side, roll dough into a log. To seal, pinch dough edge to log. Cut the log into nine slices approximately 1 1/3-inch-thick.

8. For each batch, divide 1 cup topping among nine cups of a large (1 cup capacity) muffin tin, or cover the bottom of an 8-inch square baking pan. Transfer the buns to a tin or pan.

9. Preheat oven to 375 degrees, with rack on lower level. Set a wire rack over a baking sheet; set aside. Let buns rise in a warm place until doubled in bulk, about 30 minutes. Bake buns, rotating pan halfway through, until golden brown and bubbling, 30 to 35 minutes. Immediately invert buns onto rack. Serve warm.