In a large mixing bowl, dissolve the yeast in the warm water and roughly ½ teaspoon of the sugar.
Once the yeast mixture is foamy, stir the remaining sugar, olive oil, eggs, salt, and 2 cups of flour together with a silicone spatula or wooden spoon.
When the mixture starts to come together, add the last 2 cups of flour and stir until a sticky dough forms.
NOTE - Since this is a batter-type dough, it will not be in a ball like traditional bread dough or other yeast doughs.
Cover the bowl with plastic wrap or kitchen towel. Let dough rise in a warm place until double, (about an hour).
Once the dough is risen, melt ½ cup butter in a small bowl.
Spray a baking sheet with non-stick cooking spray or line it with parchment paper.
Dump the risen dough out onto a lightly floured surface.
Bring the dough together in a ball and roll out to about ¼ inch thick.
Using a 2-inch biscuit cutter, cut the dough into circles.
Gently stretch the circles of dough into ovals.
Dip the bottom half of the oval in the melted butter. (See tip below)
Fold the ovals in half while placing them on the prepared pan, allowing the top layer to overhang the bottom fold slightly so the rolls stay shut when they bake.
Place in rows on the prepared baking sheet.
Cover the rolls with a kitchen towel for the second rise and let rise in a warm spot that's draft-free until doubled about 20 minutes.
While they're rising, preheat the oven to 350 degrees.
Bake for 20-25 minutes. The tops will be golden brown.
You can brush the tops of the rolls with any remaining butter you may have had from dipping.
Serve the warm rolls with honey butter or homemade jam!