Stir the yeast and 1 teaspoon of the sugar into the warm milk and let it sit for 5 minutes until foamy.
Beat the butter, remaining sugar, egg, and salt until smooth, then mix in the pumpkin puree and foamy yeast. The mixture will look slightly curdled, which is normal.
Add the flour and pumpkin pie spice gradually, kneading until the dough pulls away from the bowl and feels soft and only slightly tacky.
First Rise
Place the dough in a greased bowl, cover, and let rise in a warm spot for about 1 hour, until doubled.
Make the Pecan-Caramel Pan Topping
Melt the butter, brown sugar, maple syrup, and salt over medium heat, stirring until smooth, about 3 to 4 minutes.
Stir in the pecans, then pour into a greased 9x13-inch baking dish and spread evenly across the bottom.
Mix the Pumpkin-Spice Filling
Beat the softened butter, brown sugar, cinnamon, and pumpkin pie spice into a spreadable paste.
Roll, Fill, and Shape the Buns
Punch down the dough and roll it into a 24x18-inch rectangle on a lightly floured surface.
Spread the filling to the edges, leaving a 1-inch margin along one long side. Roll up tightly from the covered edge and pinch the seam to seal.
Slice into 12 even rolls and nestle them cut-side up into the caramel in the pan.
Second Rise and Bake
Cover and let the rolls rise for 45 minutes, just until puffy.
Bake at 350°F for 35 to 40 minutes, until golden and set through the center.
Invert and Cool
Let the pan rest 10 minutes, then invert onto a platter and leave it upside down for another 5 minutes so the glaze drips over the tops. Lift the pan away and serve.
Notes
Add flour to feel: Pumpkin varies in moisture, so stop once the dough is soft and slightly tacky.
Rest before flipping: The full 10-minute rest sets the caramel glossy instead of runny.
Make ahead: Shape the buns, refrigerate overnight, and bake from cold in the morning, adding a few extra minutes.
Use pure pumpkin: Not pie filling, which is pre-sweetened and spiced.