Combine the milk and water and heat until very warm (110ºF).  Pour into the bowl of a stand mixer.  Add the sugar, yeast and 1 cup of the flour.  Mix until combined but still lumpy.  Let the mixture stand until it is very bubbly, about 30 minutes.

Add the butter, salt and 2 more cups of flour to the mixture  Mix on medium speed until smooth and elastic, about 3 minutes.  Mix in 2 of the eggs until combined.  Scrape the bottom and sides of the mixer bowl as needed.  Continue to add flour until the dough is smooth.  The dough should be just slightly sticky.

Lightly oil a large bowl.  Scrape the dough out into the bowl, the turn the dough to coat in the oil.  Cover the bowl with plastic wrap and place in a warm area.  Let the dough rise until doubled, about 2 hours.

Line 2 baking sheets with parchment paper or a silpat liner.  Lightly flour a counter, then turn the dough out.  Divide the dough in half, and set one half aside.  Roll the dough into a 12-inch round circle, about 1/4-inch thick.  Brush the dough with half of the remaining beaten egg.  Sprinkle on half of the parmesan, havarti and cheddar cheese, leaving a 1-inch border around the outside.  Lightly press the cheese into the dough with your hands.

Use a pizza cutter to cut the circle into 12 wedges.  Starting on the wide end, roll each wedge up.  Place the rolls on a prepared baking sheet, making sure the end is tucked under on the bottom.  Lightly spray a piece of plastic wrap and lay over the top of the baking sheet.   Repeat with the second half of ingredients.

Let the rolls stand until they are puffy, about 45 minutes.

Preheat the oven to 375ºF, placing the racks in the upper third and lower third of the oven.  Remove the plastic wrap from the rolls and bake, switching and rotating the baking sheets halfway through, until the rolls are golden brown, about 20-25 minutes.  Cool the rolls on the baking sheets for 5 minutes before removing to a cooling rack. If desired, brush hot rolls with a bit of melted butter.