Spray your chosen baking pan(s) with cooking spray. Preheat the oven to 375°F (190°C) only after the dough has risen (see step 11).
In a large mixer bowl whisk together 2 cups of the all-purpose flour (from the 3-¼ to 3-¾ cups), ¼ cup granulated sugar, the packet of instant yeast (2¼ tsp), and 1 teaspoon salt until blended.
In a microwave-safe bowl or measuring cup combine ¾ cup milk, ¼ cup water, and ¼ cup (57 g) unsalted butter (the butter listed as "sliced into tablespoons" for the dough). Heat on HIGH in 15-second increments until the mixture is very warm but not hot to the touch (about 120°–130°F). The butter may not melt completely.
Add the warm milk mixture and 1 large egg to the flour mixture.
Using a stand mixer fitted with a dough hook (or an electric mixer or by hand), beat at medium speed for 2 minutes, scraping the bowl occasionally.
Add another 1 cup flour and beat at high speed for 2 minutes, scraping the bowl occasionally.
Add just enough of the remaining flour (from the 3-¼ to 3-¾ cup total) a little at a time so the dough forms into a ball and pulls away from the sides of the bowl. Do not exceed the total 3-¾ cups listed.
Turn the dough onto a lightly floured surface and knead until smooth and elastic and the dough springs back when lightly pressed with two fingers, about 6 to 8 minutes.
Cover the dough with a towel and let it rest for 10 minutes.
While the dough rests, make the two butter-sugar mixtures: a. For coating the pieces: melt ½ cup (113 g) unsalted butter and place it in a small bowl. In a separate small bowl combine 1 cup granulated sugar and 1 tablespoon ground cinnamon; set both bowls aside. b. For the topping: measure ½ cup (100 g) packed brown sugar, ¼ cup (57 g) unsalted butter, and 1 teaspoon ground cinnamon into a bowl; you will melt and combine these after the dough has risen.
Cut the rested dough into 36 equal pieces. (Use a bench scraper or knife: first divide into halves, then into smaller portions until you have 36 pieces, or portion to approximate equal sizes.)
Working one piece at a time, dip a dough piece into the melted ½ cup butter, then roll it in the granulated sugar + 1 tablespoon cinnamon mixture to coat. Place each coated piece into the prepared pan. Arrange pieces according to the pan option you are using: - 9×13-inch baking dish: place the pieces in a single layer. - Fluted tube pan or Bundt pan: place pieces into the pan, they will form a mound. - Two 9×5-inch loaf pans: distribute pieces evenly between the two pans. - Muffin pan for 12 monkey bread muffins: place 3 pieces in each greased muffin cup.
Cover the pan(s) with a towel and let the dough pieces rise in a warm place until doubled in size, about 45 minutes.
While the dough is rising (or just after it has doubled), preheat the oven to 375°F.
Make the topping: melt the reserved ¼ cup (57 g) unsalted butter (microwave or stovetop) and stir in the ½ cup packed brown sugar and 1 teaspoon ground cinnamon until combined and smooth. Pour this topping mixture evenly over the risen dough pieces in the pan.
Bake at 375°F according to your pan: - 9×13-inch baking dish: bake about 20 minutes, or until golden and cooked through. - Fluted tube pan or Bundt pan: bake about 40–50 minutes, or until golden and cooked through. - Two 9×5-inch bread pans: bake about 20 minutes, or until golden and cooked through. - 12 monkey bread muffins (muffin pan): bake about 20 minutes, or until golden and cooked through.
Remove from the oven when golden and cooked through. Let cool briefly before serving.