Put the water into a large mixing bowl or the bowl of a stand mixer and stir in the yeast and 1 teaspoon of the sugar with a wooden spoon. When the yeast is foamy, in about 10 minutes, add the remaining sugar, the butter and eggs.
If making the dough by hand, gradually stir in 3 cups of the flour. Beat vigorously with the spoon for several minutes until the dough becomes quite elastic (if you lift a spoonful of dough up from the bowl it should stretch for about 1 foot). Stir in another 3/4 cup flour to make a thick dough. Sprinkle the remaining 1/4 cup flour on your work surface and scrape the dough onto it. Knead about 5 minutes to incorporate the flour and to make a very smooth, soft, but supple dough that may be only slightly sticky.
If making the dough with a stand mixer, attach the flat beater and add 3 cups of flour to the mixing bowl; beat it in on low speed. Increase the speed to medium and beat for 5 minutes, until the dough begins to mass on the beater and is very stretchy. Scrape the bowl and beater and attach the dough hook. Add 3/4 cup more flour and knead on low speed for about 5 minutes, until the dough is smooth, soft, elastic, and a bit sticky. Sprinkle 2 tablespoons of the remaining 1/4 cup flour on your work surface and scrape the dough onto it. Knead about 2 minutes to incorporate the flour and to make a very smooth, soft, but supple dough that may be only slightly tacky. If too wet, knead in the last of the flour.
Wash and dry the bowl you used and either rub it with vegetable oil or coat it with cooking spray. Shape the dough into a ball and place it in the bowl. Turn to coat all surfaces with the oil or spray the top of the dough with cooking spray. Cover the bowl tightly with plastic wrap and let the dough rise at room temperature until almost tripled in size, about 2 hours.
Place the dough onto your work surface and pat it gently to deflate it. Don’t punch it. Divide the dough in half.
Roll one piece of dough into a “snake” and pinch off 7 equal pieces. Shape into round rolls and arrange in an ungreased 9-inch round pie tin. Repeat with the remaining dough. Cover loosely with a kitchen towel and let rise at room temperature until puffy and light and doubled in size, about 1 hour.
Brush with egg glaze and bake in the center of a preheated 400 degree oven for about 25 to 30 minutes, until rolls are golden brown and cooked through. Bottoms of rolls should be brown.