In the bowl of a stand mixer combine the water, yeast, and 1 tablespoon of sugar; let rest 10 minutes until foamy.
Add remaining sugar, milk, egg, salt, and enough flour to make a soft dough. Mix with the dough hook on medium-low until thoroughly combined and smooth, about 8-10 minutes. Transfer the dough to a lightly oiled bowl, cover and let rise until doubled in size, about 1 hour.
Punch down dough and knead in garlic until combined. Pinch off small handfuls of dough, about the size of a golf ball. Roll into balls, and place on a rimmed baking sheet. Cover with a towel, and allow to rise until doubled in size, about 30 minutes.
Preheat grill to high heat and lightly oil grate. Roll one ball of dough out into a thin circle. Place dough on grill, and cook for 2 to 3 minutes, or until puffy and lightly browned. Brush uncooked side with butter, and turn over. Brush cooked side with butter, and cook until browned, another 2 to 4 minutes. Remove from grill, and continue the process until all the naan has been prepare. Serve warm.