Mix warm milk, sugar, and yeast. Let it sit.
Add egg, melted butter, and salt to yeast mix.
Slowly add flour to form a soft dough.
Knead dough until smooth and soft, about 7 minutes.
Cover bowl with towel. Let it rise until doubled.
Punch dough down and divide into equal portions.
Flatten each dough piece and add shredded cheese.
Seal the dough and shape into smooth round buns.
Place buns on a baking sheet with parchment paper.
Cover and let buns rise again for 30 minutes.
Preheat oven to 350°F while buns rise fully.
Brush buns with egg yolk mixed with milk gently.
Sprinkle extra cheese on top before baking if desired.
Bake for 18-22 minutes until golden and puffed.
Cool slightly before serving your baked cheese buns recipe.