Combine warm water and milk in a mixing bowl and sprinkle yeast over. Let sit until frothy, about 10 minutes.

Add sugar, egg, and melted butter for dough to the yeast mixture. Mix with a whisk before adding 75% of the flour with the salt. Mix, adding more flour, until a very soft and sticky dough is formed. Let knead in the mixer for about 5 minutes. Cover and let rise in a warm spot until doubled in size, about 1 hour.

While dough is rising, preheat the oven to 375 degrees F (190 degrees C). Generously butter a 9x13-inch metal baking pan.

Combine brown sugar, white sugar, salt, melted butter, and water for topping in a bowl. Mix thoroughly until smooth. Pour into the prepared pan and spread evenly to cover the bottom. Scatter pecans evenly over the top. Set aside until needed.

Combine brown sugar and cinnamon in a bowl for filling; mix until thoroughly combined. Set aside until needed.

Transfer dough onto a lightly floured surface. Lightly flour your hands and press and stretch the dough to form a 18x15-inch rectangle. Sprinkle cinnamon-sugar mixture evenly over the dough to the edges, leaving a 2-inch border along the edges. Lightly press the sugar mixture into the dough with your hands.

Roll the dough into a cylinder with lightly floured hands starting with the edge closest to you; try not to roll too tightly. Finish shaping the cylinder as uniformly as possible, seam-side down. Lightly score the roll with the edge of a knife to indicate 12 equal portions.

Slide a piece of string or floss under the dough, lining it up at the first knife mark. Cross the ends of the string over the top and pull in opposite directions to cut through the dough. Continue with remaining dough.

Transfer buns into the pan with topping, making 3 rows of 4 buns. If one side of a bun has more dough than another, place with the doughier side up in the pan. Tent the pan loosely with foil and let rise until buns have almost doubled in size, 45 minutes to 1 hour. Save foil in case you need it towards the end of baking time.

Bake in the preheated oven until an instant-read thermometer inserted into the center of a bun reaches 200 degrees F (93 degrees C), about 35 minutes. If the tops are getting too browned, loosely tent the pan with foil for the last 5 to 10 minutes of baking time.

Remove from the oven onto a wire rack and let cool for 5 minutes. Turn pan over carefully onto a serving platter. Use a spoon to transfer any sticky topping that has remained in the pan. Let cool and serve warm or at room temperature.