Cream together the sugar, butter, and salt on medium-high speed in an electric mixer with a paddle attachment. Whip in the egg and vanilla extract until smooth. Then add the flour, yeast, and milk. Mix on low speed until the dough forms a ball. Switch to the dough hook and increase the speed to medium, mixing for approximately 10 minutes (or knead by hand for 12 to 15 minutes), or until the dough is silky and supple, tacky but not sticky. You may have to add a little flour or water while mixing to achieve this texture. Lightly oil a large bowl and transfer the dough to the bowl, rolling it around to coat it with oil. Cover the bowl with plastic wrap. Let the dough rise at room temperature for approximately 2 hours, or until the dough doubles in size.
Meanwhile, make the Caramel Glaze: In the bowl of an electric mixer, combine the butter, granulated sugar, brown sugar, and salt, and cream together for 2 minutes on high speed with the paddle attachment. Add the corn syrup and vanilla extract, and continue to cream for about 5 minutes, or until light and fluffy.
Transfer the dough to the counter. Roll out the dough with a rolling pin, lightly dusting the top of the dough with flour to keep it from sticking to the pin. Roll it into a rectangle 18 inches wide by 9 inches long. Sprinkle the cinnamon sugar over the surface of the dough (use it all!), and roll the dough up with the long side facing you, creating a cinnamon-sugar spiral as you roll. With the seam side down, cut the dough into 12 even pieces.
Coat the bottom of a 9x13-inch baking dish with the caramel glaze, then sprinkle the pecans evenly over the surface. Lay the pieces of dough cut-side-up on top of the caramel glaze, evenly spaced throughout the pan. Cover loosely with plastic wrap and allow to rise at room temperature for 75 to 90 minutes, or until the pieces have grown into one another and have nearly doubled in size.
Make-Ahead Note: Instead of the second rise, you can place the shaped buns in the refrigerate for up to 2 days, pulling the pan out of the refrigerator 3 to 4 hours before baking to allow the dough to proof at room temperature.
Preheat the oven to 350 degrees F and adjust the oven rack to the lowest shelf.
Bake the sticky buns for 30 to 40 minutes, or until golden brown. Cool the buns in the pan for 5 to 10 minutes, and then remove them by flipping them over onto another pan or serving platter. Carefully scoop any run-off glaze back over the buns with a spatula. Wait at least 20 minutes before serving. The sticky buns are best eaten the day of baking but can be stored, covered, at room temperature for up to 2 days (see notes below on reheating).