In a large bowl or the bowl of a stand mixer, combine the warm milk, yeast, and one tablespoon of the granulated sugar. Let it sit for about 5 to 10 minutes until it becomes foamy. Add the remaining granulated sugar, melted butter, egg, salt, and half of the flour. Mix until combined, then gradually add the remaining flour until a soft dough forms.
Knead the dough on a lightly floured surface for about 5-7 minutes, or use a dough hook on medium-low speed for 5 minutes. The dough should be smooth and slightly tacky. Place the dough in a greased bowl, cover with a clean towel, and let it rise in a warm, draft-free spot for 1 to 1.5 hours, or until it has doubled in size.
While the dough is rising, prepare your baking pan. In a small bowl, whisk together the 1/4 cup melted butter, maple syrup, and 1/2 cup of the brown sugar. Pour this mixture into the bottom of a 9×13 inch baking pan, spreading it evenly. Sprinkle the chopped pecans over the top of the syrup mixture.
Once the dough has doubled, punch it down and turn it out onto a floured surface. Roll it into a large rectangle, about 12×18 inches. Spread the 1/2 cup of softened butter over the dough. In a small bowl, mix the remaining 1/2 cup brown sugar and cinnamon, then sprinkle it evenly over the buttered dough.
Starting from the long edge, roll the dough up tightly into a log. Cut the log into 12 even slices using a sharp knife or dental floss. Place the rolls cut-side down in the prepared baking pan on top of the pecan mixture. Cover and let them rise for another 45 minutes until puffy.
Preheat your oven to 350°F (175°C). Bake the buns for 25 to 30 minutes, or until they are a deep golden brown. If they seem to be browning too quickly, you can tent them loosely with aluminum foil for the last 10 minutes of baking.
Remove the pan from the oven and let it cool for only 5 minutes. Place a large serving platter or a sheet of parchment paper over the pan and carefully invert it. Let the glaze drip down over the buns for a minute before removing the pan. Serve warm.