Prepare Bread Pudding: Whisk together eggs and next 8 ingredients in a large bowl until well blended. Add bread pieces, stirring to thoroughly coat. Cover with plastic wrap, and chill 8 to 24 hours.
Preheat oven to 350°. Spoon chilled bread mixture into a lightly greased 13- x 9-inch baking dish. Cover with aluminum foil.
Bake, covered, at 350° for 35 minutes. Uncover and bake 15 minutes.
During last 15 minutes of baking, prepare Caramel-Pecan Sauce: Heat pecans in a medium skillet over medium-low heat, stirring often, 3 to 5 minutes or until lightly toasted and fragrant.
Cook brown sugar, butter, and corn syrup in a small saucepan over medium heat, stirring occasionally, 3 to 4 minutes or until sugar is dissolved. Remove from heat; stir in vanilla and pecans.
Remove bread pudding from oven; drizzle with Caramel-Pecan Sauce. Bake 5 minutes or until sauce is thoroughly heated and begins to bubble.
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