FOR THE DOUGH:
Years ago, Caroline Craig saw a recipe in the local newspaper for bread made with leftover winter squash. She tried using it to make her sticky buns; the result was so pleasing that she's made them that way ever since.
Note: If you don't happen to have any leftover winter squash on hand, you can make this recipe with canned squash or pumpkin. Because canned squash has more moisture, you'll need more flour, closer to the 4 cups listed below. But the buns still turned out beautifully in our tests.
First, make the dough; Put squash and hot water into the bowl of a standing mixer or the jar of a blender, and beat (or whirl) until smooth. If you're using a blender, transfer mixture to a large bowl; otherwise, leave it in your mixer's bowl. Add sugar and 1-½ teaspoons yeast; then stir and let mixture sit 10 minutes (this will activate the yeast). Add salt, melted butter, and remaining ½ teaspoon yeast. Whisk until blended.
Beat in 2 cups flour. Keep adding flour, ¼ cup at a time, until dough becomes workable without being too sticky to touch; the amount you'll need will depend on the type of squash (and flour) you use. Don't worry–you'll be able to feel it when the dough ceases to be too sticky. If you're mixing by hand, turn dough out onto a well-floured surface to work in the additional flour. If you're using a standing mixer, switch to the dough-hook attachment.
Continue kneading until dough springs back to the touch and feels very smooth, 5 to 10 minutes. Then coat dough with oil, return it to the bowl, cover with a kitchen towel or plastic wrap, and place in a warm spot until doubled in size, 1 to 1-½ hours.
While dough is rising, butter a 9x9-inch baking dish or a standard 12-cup muffin tin. Set aside.
FOR THE TOPPING:
Next, make the topping: In a small saucepan, mix brown sugar, butter, and corn syrup together. Set over medium-high heat and cook, stirring, until brown sugar melts and mixture is bubbling. Remove sauce from heat and pour into the bottom of the prepared baking dish (or, if you're using a muffin tin, put about 1 tablespoon sauce into each well). Sprinkle with nuts and set aside.
FOR THE FILLING:
Now make the filling: In a small bowl, stir together melted butter, sugar, and cinnamon. Set aside.
When dough is ready, dust your work surface with flour and roll dough out into a rectangle, with the long side facing you. Spread filling all over dough, leaving a 1-inch border at the top (this will be your seam). Roll dough up from the bottom, jellyroll style, and press the seam to seal it. With a sharp knife, cut the log into 9 slices (if using a baking dish) or 12 slices (if using the muffin tin). Transfer to the prepared dish, with cut sides facing up. Cover and set to rise again until doubled, about 45 minutes.
About 15 minutes before you're ready to bake, preheat oven to 350° and set the oven rack to the middle position. Bake until tops are nicely browned and syrup is bubbling, 30 to 40 minutes. Serve warm.