In the bowl of an electric mixer, add the warm water (approx 115 degrees), sugar and yeast. Stir until all the yeast has dissolved and let sit for about 10 minutes, or until it starts to get bubbly.
Add the warm milk, eggs, salt and 2 cups of flour to the yeast mixture, then mix with a bread hook attachment on low for 1 minute. Add the remaining 4 - 4 ½ cups of flour slowly, adding about ½ cup of flour at a time until the dough pulls away from the side of the bowl.
When the dough pulls away from the side of the bowl, stop adding flour and continue to mix on low for about 2 minutes. Add the shredded cheese and mix on low until it's fully incorporated. If it starts sticking again, add a little more flour.
Cover the mixing bowl with a towel and let the dough rise until double, or about 30 minutes.
Preheat oven to 350 degrees F.
Once the dough has doubled in size, divide in two equal pieces and shape into round loaves by gently rolling and tucking the edges under. Place the shaped dough balls onto a sheet pan, cover loosely with a kitchen towel and let the dough rise again until double, or about 30 minutes.
After the dough has doubled in size, score the top of the cheddar cheese loaves with a knife or scoring tool. In a small bowl, add the egg and water and whisk to combine, then brush the egg wash lightly over the top and sides of each loaf.
Bake in a preheated degree oven for 30-40 minutes, or until golden brown on top. If the bread begins to brown too quickly, cover the loaves loosely with aluminum foil and continue to bake until 40 minutes is up.
Remove from the oven and transfer the loaves to a wire cooling rack to cool completely before slicing and serving. Enjoy!