Preheat the oven to 350 degrees F. Spread the pecans on a baking sheet and bake until lightly toasted, about 8 minutes; set aside.
Melt 1 stick butter in a medium saucepan over medium heat.
Add the brown sugar and heavy cream; boil until slightly thickened, about 4 minutes.
Stir in the vanilla, vinegar, rum, salt and the toasted pecans.
Pour all but 1/2 cup of the pecan mixture into an 8-inch-round, 3-inch-deep cake pan.
Refrigerate until almost set, about 15 minutes. Set the remaining pecan mixture aside at room temperature.
Put the remaining 1/2 stick butter in a microwave-safe bowl; microwave until melted.
Put the granulated sugar in another bowl.
Pinch off tablespoonfuls of dough and roll into 1-inch balls.
Dip the balls in the melted butter, letting the excess drip off, then roll in the granulated sugar; arrange in the prepared cake pan in 2 layers, leaving a bit of space between the balls.
Let rise, uncovered, in a warm place until doubled in size, about 1 hour, 30 minutes.
Preheat the oven to 350 degrees F.
Bake the bread until puffed and golden brown (a thermometer inserted into the center should register 175 degrees F), about 55 minutes; tent with aluminum foil if the top is getting too dark.
Remove the bread from the oven and spread the reserved 1/2 cup pecan mixture on top. Transfer to a rack and let cool 10 minutes in the pan, then invert onto a plate. Pull off pieces to serve.