To make the dough, put the warm water in a large bowl (or the bowl of your stand mixer) and sprinkle with the yeast and a pinch of the sugar. Let stand 5 minutes, or until foamy. (If it doesn’t foam, toss it and buy fresh yeast.)
In a small bowl, mix the warm milk and eggs together with a fork. Add to the yeast mixture along with 3 cups of the flour and the remaining sugar; mix until well blended and sticky. Add the butter and remaining flour and stir or beat with the dough hook attachment of your stand mixer until you have a soft, sticky dough; knead for about 8 minutes, until smooth and elastic. It will still be slightly tacky. Place back in the bowl, cover with a tea towel and let rise in a warm place for an hour, or until doubled in bulk. Melt the butter in a small dish and add the garlic; set it aside to infuse the butter.
To make the buns, divide the dough in half, shape each into a rough square or rectangle (this will make it roll out more evenly) and on a lightly floured surface, roll into a rectangle that’s about 10×15-inches (or slightly bigger, even) and about 1/4-inch thick. Brush each piece with half the melted butter (including the chunks of garlic - or not) and scatter with grated cheese.
Starting on a long side, roll the dough up into a log, and using a serrated knife, cut it crosswise into thirds. Cut each piece into thirds – this is easier than eyeballing it to get 9 even pieces. Place cut-side-up into two 8 or 9-inch round pans, placing one in the middle and the rest around it, or in an 8x8-inch or 9x9-inch square pan in 3 rows of 3. Cover with a tea towel and let rise for another hour, until doubled in bulk. (If you’re making them the night before, cover and place in the fridge for a slow rise; take them out and leave them on the countertop for 1/2 hour or so before baking.)
When you’re ready to bake, preheat the oven to 350F. Bke for 30-40 minutes, until deep golden. Let cool for 5-10 minutes, but invert onto a plate while still warm. Seve warm.
Serving Size Makes 1 1/2 dozen garlic-cheese buns.