Combine the water, milk, 1 tablespoon of the melted butter, eggs, sugar, salt, and yeast in the bowl of a stand mixer. Add 2 cups of the flour and stir with a wooden spoon until the dough forms a rough, shaggy mass. Attach the dough hook to the mixer, turn to medium-low speed, and gradually add the remaining flour, kneading until a mass of dough begins to form. Continue kneading on medium-high speed for 4 to 5 minutes until a soft, smooth ball of dough is formed. The dough should feel elastic and slightly tacky to the touch.

Lightly spray a large clean bowl with cooking spray and place the dough in the bowl. Cover the bowl lightly with plastic wrap. Let rise for about 1 hour to 90 minutes at room temperature or until the dough is big, puffy, and about doubled in size.

Grease a 9x13-inch or similarly shaped baking pan with butter. Gently deflate the dough. Use a bench scraper, knife, or pizza wheel to divide the dough into 12 equal pieces. Shape each piece into a ball and place in the prepared pan. Lightly cover the dough with plastic wrap and let the rolls rise for about 1 hour, or until about doubled in size.

Preheat the oven to 375°F.

Brush the rolls gently with the beaten egg. Bake the rolls for 20 minutes, or until golden brown. Brush with melted butter and sprinkle with salt. Serve warm. Store leftovers in an airtight container or ziptop bag for up to 3 days.