Whisk the warm milk, dry active yeast, and ¼ tsp of sugar in a small cup. Let it sit for about 5-8 minutes so the yeast can activate. Once it is foamy on top, you can add the melted butter and stir.
Whisk the flour, sugar, and salt in the bowl of your stand mixer. Add the yeast mixture and knead with the dough hook attachment for about 8-10 minutes or until the dough is smooth and elastic. You can also mix the dry ingredients with the yeast mixture in a large bowl. Then, knead the dough by hand on a lightly floured surface for about 8-10 minutes.
Grease a large bowl with the oil. Place the dough in the bowl, turning it to cover all sides with the oil. Cover the bowl with plastic wrap or a clean kitchen towel and let it rest in a warm place for about 1-2 hours, or until it doubles in size.
Once the dough has doubled in size, you can shape it into a log and divide it into 8 small pieces. Flatten each piece with a rolling pin until it is about 7-8 inches (17-20 cm) long. Sprinkle the inside of the dough with shredded mozzarella cheese to taste, if using, and roll the dough starting at the top to form a small loaf. Another option is placing a small piece of string cheese on the top edge of the dough, and rolling it down to form a small loaf. Place all shaped loaves on a greased baking sheet, loosely cover them with plastic wrap or a clean kitchen towel, and let them rest a second time for about 45 minutes to 1 hour or until they grow in size.
Preheat the oven to 350°F (180°C). Brush each loaf with the egg wash and sprinkle the tops with the Everything Bagel Seasoning or your favorite seasoning, to taste. Bake for about 30-35 minutes or until golden brown on top.