Prep the loaf pan. Spray a 9 x 5-inch loaf pan with cooking spray and line it with parchment paper, leaving a 2-inch overhang on the long sides. Spray the parchment, too.
Roll out the dough. Remove the dough from the refrigerator, uncover it, and discard the plastic wrap. Tip it onto a lightly floured counter and use a rolling pin to roll the dough into a rough oval about ½ inch thick. Make the dough balls. Use a bench scraper or pastry cutter to divide the dough into 1-inch squares. Working quickly, use your hands to roll each square as you would clay or Play-Doh, rolling it back and forth in quick motions until it forms a small ball. Place the balls in a large bowl as you go. Coat the dough balls. Drizzle the dough balls with the melted butter for the coating and gently toss until the dough balls are completely coated. Whisk together the brown sugar, granulated sugar, and cinnamon for the coating in a small bowl. Sprinkle the mixture over the dough balls and toss until the dough balls are completely coated.
Prep the dough for proofing. Place the dough balls in the prepared pan. Cover with plastic wrap and let sit in a warm spot for 1 to 2 hours, or until puffy.
Preheat the oven and the sheet pan. About 30 minutes into the second rise, position a rack in the center of the oven and preheat the oven to 350°F. Line a quarter or half sheet pan with parchment paper.
Make the caramel sauce. When the dough is ready for baking, make the caramel sauce. Melt the butter for the caramel sauce in a small, heavy-bottomed saucepan. Remove from the heat, immediately add the sugar, salt, and vanilla, and whisk until smooth.
Finish assembling the monkey bread for baking. Place the loaf pan with the dough on top of the prepared sheet pan. Pour the warm caramel over the dough, coating the top of the dough completely.
Bake the monkey bread. Bake the monkey bread for 40 to 45 minutes, or until the top of the loaf is golden brown. An instant read thermometer inserted into the center of the loaf should read between 190°F and 200°F. Serve and store. Serve immediately, or warmed. The monkey bread can be stored at room temperature, tightly wrapped in plastic wrap, for up to 3 days.