Prepare a heavy-duty stand mixer with kneader blade attachment.
In a small saucepan heat the milk with butter and salt (the sugar will be proofed with the yeast) until the milk is warmed and the butter is melted; remove from stove top and cool to lukewarm (I stick the saucepan in the fridge for about 20 minutes).
Proof 1 tablespoon plus 1 teaspoon dry yeast in 1-1/2 cups warm water with 2 tablespoons sugar until foamy (about 8-10 minutes).
Place 4-1/2 cups flour in the stainless steel mixing bowl.
When the milk mixture is luke warm and the yeast is proofed add to the mixing bowl.
Start kneading adding in more flour if needed to create a soft smooth dough; knead for about 8 minutes.
The last couple of minutes of kneading add in the grated cheddar cheese; mix until combined with the dough.
Remove the dough to the counter; cover with a clean towel and let rest about 5 minutes.
Gather up the dough and shape into a ball.
Place in an oiled bowl; cover with plastic wrap or a clean towel, then let rise in a warm place for about 1 hour.
Punch down the dough and let rest covered for about 5-6 minutes.
Divide the dough evenly in half and shape into two oval loaves.
Place the two loaves onto a large greased baking sheet (leaving room between the dough).
Cover and let rise for about 30 minutes.
In a small bowl mix together the egg and water; brush liberally onto the top and sides of the dough (you will have some egg mixture leftover).
Bake in a 400 degree oven for about 23 minutes or until lighly golden brown; remov quickly sprinkle with grated cheese, place back into the oven and bake until browned (another 3-5 minutes).