Special equipment: a 9-inch round baking dish
Preheat the oven to 375 degrees F.
In a pot set over medium-low heat, melt the butter. Add 3/4 cup of the brown sugar, the vanilla and 1/4 cup water, and cook until the sugar dissolves. Spread 3 tablespoons of the sugar mixture on the bottom of a 9-inch round baking dish, and sprinkle the 1/2 cup whole pecans over the sugar mixture.
Dust a clean surface with flour, and roll out the dough to a 10-by-14-inch rectangle. Cover the dough with the remaining 1/2 cup of the brown sugar, the 1/2 cup chopped pecans and the remaining sugar mixture. Starting at one of the long ends, gently roll the dough into a log. Gently pinch the seam with your fingers to seal, and then cut the log into 12 equal pieces. Place the dough in the baking dish and bake until golden brown, 30 minutes.
Rest the sticky buns for 5 minutes. Place a flat serving platter over the baking dish and flip to remove the buns. Serve warm.