4 1/2 to 5 cups unbleached all-purpose flour, plus more for dusting and
1 tablespoon vegetable oil
4 tablespoons unsalted butter, melted
In a large bowl, mix the yeast with 1 tablespoon of the sugar and 1/4 cup of the water and let stand until foamy, 5 to 10 minutes. Stir in the egg, milk, salt and the remaining 1/4 cup of sugar and 3/4 cup of water. Add 4 cups of the flour and stir to form a dough that is soft and pliable but not sticky. Knead the dough until smooth and elastic, adding more flour if it seems very tacky. You can do this by hand on a floured work surface, in a food processor or in a standing electric mixer fitted with the dough hook.
Rub 1/2 tablespoon of the oil in a large bowl. Transfer the dough to the bowl and brush the top with the remaining 1/2 tablespoon of oil. Cover with a kitchen towel and let rise in a warm, draft-free spot until doubled in bulk, 1 to 1 1/2 hours.
Punch down the dough and pinch off 2-inch pieces; roll them into smooth balls. Place the balls on a lightly floured baking sheet and cover with a lightly dampened kitchen towel. Let rise again until puffy, about 30 minutes.
Light the grill. When it's hot, place a rolling pin, cutting board, bowl of flour, the melted butter and a brush near the grill. On a lightly floured cutting board, roll out a ball of dough to a 5-inch round. Gently slap the dough from one hand to the other to stretch it into a 7-to 8-inch oval. Stretch the oval into a traditional teardrop shape and immediately lay it on the hot grate. Grill the naan until the bottom is crusty and browned and the top is puffed and blistered, about 40 seconds. Brush with butter. Invert the naan and grill the other side until lightly browned, about 30 seconds longer. Grill the remaining naan the same way. Brush each naan with more butter as it comes off the grill and serve piping hot. Serve whole, or cut each naan into 3 wedges to serve the traditional way.