In a bowl, pour 1/4 cup rum over raisins; let stand until slightly plumped, at least 1 hour. Butter a 9-inch square springform or regular baking pan.
In a food processor, whirl 1 cup pecans, 1/4 cup butter, 1/4 cup brown sugar, cinnamon, salt, and raisins with rum until finely chopped.
On a floured surface, with a floured rolling pin, roll bread dough into a 16- by 8-inch rectangle. (If dough springs back, let rest for 5 minutes, then roll again.) Spread rum-raisin filling over dough, leaving a 1/2-inch margin bare along one long edge.
Starting at opposite edge, roll dough around filling into a cylinder. Brush egg mixture over bare edge, then pinch against cylinder to seal. Cut off ragged ends, then cut cylinder into 16 equal rounds. Space rolls evenly, each on a cut side, in pan. Cover with plastic wrap and let stand until doubled, about 40 minutes.
Uncover and brush rolls lightly with egg mixture. Bake in a 350º regular or convection oven until browned, 25 to 30 minutes. Remove pan rim (if using a regular pan, invert rolls onto a wire rack, then invert again onto a rimmed plate).
In a 1 1/2- to 2-quart pan over medium-high heat, combine remaining 3/4 cup brown sugar, 1/4 cup butter, and 1/2 cup pecans with corn syrup. Stir until mixture is thick and bubbling, 3 to 5 minutes. Stir in remaining 2 tablespoons rum. Pour and scrape evenly over warm buns. If excess glaze pools at base, spoon over tops of buns. Serve warm.
Nutritional analysis per sticky bun.