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 (254 grams) 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 ball should feel elastic and slightly tacky to the touch.
Lightly spray a large bowl with cooking spray and place the dough in the bowl. Cover the bowl lightly with plastic wrap or a kitchen towel.
Let dough 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 9×13-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 it 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 for 20 minutes or until golden brown.
Brush with melted butter and sprinkle with flaky salt. Serve warm. Store leftovers in an airtight container or zip-top bag for up to 3 days.
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Notes
Much like milk bread or a brioche, this dough is enriched with milk, butter, and sugar, which can make the dough a little sticky. Your instinct might be to add extra flour, but that will create a denser, tougher roll instead of the fluffy dinner roll you want.If you find the dough tricky to handle, dampen your hands with water. If you use instant yeast and don’t want to wait extra time for the dough to rise, you can speed the process up slightly. Combine it with the warm water and milk called for in the recipe and allow it to proof for 5 minutes before adding it to the bowl.If you’re looking for bread machine recipes, this one could work. Just add the ingredients in the order recommended by your machine, use the “dough” setting, and let it handle the mixing and first rise. Follow my instructions from there!