Combine the yeast, ½ teaspoon sugar, and ¼ cup warm water in a 2-cup liquid measuring cup. Stir just to mix and let stand 5 minutes. (If the yeast doesn’t activate (or grow), discard and try again with fresh yeast.)
4½ teaspoons active dry yeast, 1¼ cups water, ½ teaspoon granulated sugar
In a large bowl, combine the yeast mixture, remaining 1 cup warm water, 2 cups flour, eggs, melted (cooled) shortening, ½ cup sugar, and salt. Use a wooden spoon or Danish Dough Whisk to beat the mixture together for 2 minutes. Gradually stir in enough of the remaining flour until a soft dough forms. The dough shouldn’t be too sticky.
4½-5 cups all-purpose flour, 3 large eggs, ½ cup vegetable shortening, ½ cup granulated sugar, 4 teaspoons kosher salt
Spray a different, large, glass bowl or a proofing container with nonstick spray. Place the dough in the bowl and roll it once to oil all sides of the dough.
Loosely cover the bowl with sprayed plastic wrap or a damp, clean, lightweight dishtowel. Place it in a proofing oven (around 100°F) or in a warm, dry place until doubled, about 1 hour.
Punch the dough down, cover tightly, and refrigerate at least 8 hours.
About 2 hours before baking, remove the dough from the fridge, and spray 2 9-inch square pans with nonstick spray. Melt ½ cup butter.
½ cup salted butter
Punch the dough down a second time and place it on a lightly floured surface. Knead the dough 5-6 times, divide it in half, and cut each half into 16 equal portions. Shape each portion into 2-inch balls, dip them in the melted butter, and place equally apart (with a little space in-between rolls) into the 2 prepared pans.
Loosely cover each pan of rolls with sprayed plastic wrap and place is a warm, dry place for 1½ hours, or until the rolls are doubled in size.
Bake the rolls, in a preheated 375°F oven, for 12 minutes, or until golden brown. The internal temperature should register 200°F to 205°F on an instant-read thermometer, when tested in the middle of a middle roll.
While the rolls bake, cover a large metal cooling rack with a clean dishcloth.
When the rolls are ready, transfer them to the cooling rack and let cool 1 minute. Turn the rolls out onto the towel-covered cooling rack and then turn them upright. Brush the rolls with melted butter, if desired.
Serve immediately or cool completely and freeze for later use.