In a bowl, mix the yeast, sugar, and milk. Add the flour and mix until combined.
Add the salt and softened butter, then stretch and fold (or coil fold) until the butter is fully incorporated.
Cover and let the dough rise for about 2 hours, or until doubled.
Mix the garlic butter ingredients together.
Divide the dough into 4 equal portions. Flatten each one, spread 1 tbsp garlic butter, add cheese, fold it, make cuts as shown, then fold again.
Place in an oven-safe dish lined with parchment paper. Cover and let rest for 20 minutes.
Brush with milk, then bake at 380°F for 25 minutes.
Cover immediately after baking until no longer hot, then brush with the remaining garlic butter.
