Preheat the oven to 375 degrees F.
For the topping: In a pan, melt the butter and add the pecans, corn syrup and vanilla. Let the mixture bubble up for a minute, then remove from the heat. Spoon 1 teaspoon of the mixture into each cup of three 24-hole mini-muffin tins.
For the rolls: Roll out each piece of dough into a rectangle 6 inches wide and 24 inches long (should roll out very thin!). Pour a third of the melted butter on each piece, then sprinkle a third of the sugar and a third of the cinnamon evenly over each surface.
Starting at the far end of 1 piece, roll the dough into a long roll, rolling toward you. You'll end up with a 24-inch-long roll of dough. With a sharp knife, cut into very thin slices, about 1/4 inch thick. Repeat with the remaining pieces of dough.
Lay the sliced rolls into the muffin tins on top of the pecan mixture, tucking the tails under and patting them firmly on top of the topping. Bake until golden brown, 12 to 13 minutes.
Turn out the rolls from the pans immediately. Invert carefully, so that the rolls turn out and the pecans stay stuck to the bottoms (which are now the tops!). If a little of the pecans stick in the tins, just dig them out with a spoon and reapply them to the rolls.