Prepare the Topping: In a 9x13-inch baking pan, whisk together the brown sugar, melted butter, heavy cream, corn syrup, and salt until smooth. Spread the mixture evenly over the bottom of the pan. Sprinkle the chopped pecans over the caramel mixture. Set aside.
Activate the Yeast: In the bowl of a stand mixer or a large bowl, dissolve the yeast in the warm water with a pinch of sugar. Let it sit for 5-10 minutes until foamy.
Make the Dough: To the yeast mixture, add the warm milk, pumpkin puree, sugar, egg, vanilla, and salt. Whisk to combine. Add 3 1/2 cups of flour and mix with the dough hook attachment on low speed until a shaggy dough forms. Add the softened butter, one tablespoon at a time. Increase speed to medium and knead for 5-7 minutes, adding the remaining 1/2 cup of flour as needed, until the dough is smooth and elastic.
First Rise: Place the dough in a lightly oiled bowl, turning to coat. Cover with plastic wrap and let it rise in a warm place for 1.5 to 2 hours, or until doubled in size.
Assemble the Buns: Punch down the dough and turn it out onto a lightly floured surface. Roll it into a 12x18-inch rectangle. In a small bowl, mix the brown sugar and spices for the filling. Spread the 4 tablespoons of softened butter over the dough, leaving a 1/2-inch border. Sprinkle the cinnamon-sugar mixture evenly over the butter.
Roll and Cut: Starting from the long edge, roll the dough into a tight log. Pinch the seam to seal. Using a sharp knife or dental floss, cut the log into 12 equal-sized rolls (about 1.5 inches each). Arrange the rolls cut-side down in the prepared baking pan on top of the pecan caramel.
Second Rise: Cover the pan with plastic wrap and let the buns rise for another 45-60 minutes, or until puffy.
Bake: Preheat your oven to 375°F (190°C). Bake for 25-30 minutes, or until the tops are golden brown. If they brown too quickly, tent the pan with foil.
Invert and Serve: Let the pan cool for only 5 minutes. Carefully place a large serving platter or cutting board over the pan and quickly flip it over. Tap the top of the pan to release the buns and the topping. Scrape any remaining caramel and pecans from the pan onto the buns. Serve warm.