Dissolve the sugar in lukewarm water in a bowl. Mix in the yeast and stir. Let it rest for 5 minutes in a warm place in your kitchen, covered with a clean kitchen towel. At the five-minute mark, it should be foamy. If it isn't, the yeast is too old and needs to be replaced.
Step 2: Adding flour
If the yeast is OK, pour the yeast mixture into the mixer bowl, and add flour, milk, egg, melted butter, salt, and the dissolved yeast to the mixing bowl. Add 1¾ cup of cheddar and mix briefly. Cover with a clean tea towel, and let the dough rise in a warm, dark place until it doubles in volume (about an hour).
Step 3: Kneading the dough
Using the kneading attachment, knead for 10 mins (the dough will be loose and shaggy). If you don't have a mixer, you can use a silicone spatula and fold the dough in the bowl, pressing each time for 10 minutes.
Step 4: Resting the dough
Add 1¾ cup of the shredded cheese and fold a few times to incorporate it into the dough. Cover the dough with a clean tea towel, and let the dough rise in a warm, dark place until it doubles in volume (about an hour).
Step 5: Making the bread rolls
Pour the dough on a clean, floured surface. Press without kneading until it has lost all the gas. Divide into 6 balls of equal size. Sprinkle flour as needed to prevent it from sticking to your hands.Grease the muffin pan and place each ball into a hole in the muffin pan.
Step 6: Let the dough to rise
Cover with a clean tea towel, and let the dough rise in a warm, dark place until it doubles in volume (about an hour).
Step 7: Make eggwash and sprinkle cheese
Pre-heat oven to 400 ºF [200 ºC]. In the meantime, make the egg wash by whisking together the egg yolk and 1 tablespoon of milk. Use it to brush the rolls with it.Sprinkle the remaining cheese on the rolls.
Step 8: Baking
Bake the rolls for 15-20 minutes or until golden brown.Remove from the oven and serve warm.