In a medium bowl, combine 4-1/2 oz. (1 cup) of the flour with the yeast. Stir in the warm milk or water until combined. Cover the bowl and let sit in a warm spot in the kitchen for 30 minutes. Stir the banana, melted butter, sugar, egg, and salt into the yeast mixture until well combined. Stir in all but 1 oz. (1/4 cup) of the remaining flour to make a stiff, shaggy dough. Turn the dough out onto a lightly floured surface.
Knead, folding the dough over onto itself. If the dough sticks, use a dough scraper to pick up the sticky bits and sprinkle a small amount of the remaining flour onto the surface. Continue to knead until the dough becomes smooth and easy to handle, 5 to 10 minutes. Put the dough in a lightly buttered medium bowl, cover with plastic wrap, and let rise in a warm place until doubled, 40 to 50 minutes.
Fill the rolls:
Turn the dough out onto a lightly floured surface and shape into a rectangle. Roll out into a 16 x 11-inch rectangle. With an offset spatula, spread the softened butter evenly over the dough to within 1/8 inch of the edge. Combine the chopped pecans, brown sugar, and cinnamon in a small bowl; sprinkle the mixture evenly over the butter.
Roll up the dough lengthwise, starting from a long edge, and pinch the seam to seal. Turn the log so that it’s seam side down. Trim off and discard 1/2 inch from each end; then cut the log into twelve 1-1/4-inch pieces. Cover with plastic wrap.
Make the caramel-pecan sauce:
Generously butter a 12-cup standard muffin pan.
In a medium saucepan over medium-high heat, combine the sugar and 1/4 cup warm water and stir until the sugar dissolves, 2 to 3 minutes. Bring the syrup to a boil and cook, without stirring, until it begins to caramelize. Gently swirl the pan to help the syrup brown evenly. Once the syrup has turned an amber color, remove the pan from the heat and carefully stir in the banana and cold butter. Stir in the heavy cream, returning the pan to low heat if the mixture doesn’t smooth out immediately. Add the vanilla and salt and stir until smooth. Strain the sauce, discarding the banana. Let the caramel cool until just warm, about 15 minutes. Drizzle about 1-1/2 Tbs. of caramel into each muffin cup, reserving the extra sauce at room temperature. Sprinkle the pecans evenly among the cups.
Place the dough slices over the sauce and nuts, cut sides down. Cover with plastic wrap and refrigerate overnight.
Bake the sticky buns:
Remove the pan from the refrigerator and let the dough rise in a warm spot until doubled, 1 to 1-1/2 hours.
Position racks in the center and lower third of the oven and heat the oven to 350°F Put the muffin pan on the center rack and set a foil-lined baking sheet on the rack below to catch any overflowing syrup. Bake until the tops and edges of the buns are browned, 20 to 22 minutes.
Immediately invert the pan onto a rimmed baking sheet, replace any pecans that fell off, and let cool for 10 minutes. Reheat the reserved caramel sauce and serve the buns drizzled with the sauce.