Line a baking sheet with parchment paper.
Place the yeast packet into a large mixing bowl and whisk in ½ cup flour with the warm water until the mixture is smooth.
Set aside and allow the yeast to proof for 15 minutes. The mixture will have proofed successfully once you notice that it is foamy.
Whisk in the same bowl the egg, melted butter, sugar, and salt until combined and then add in the 3 cups of flour and begin to knead by turning around in the bowl until the dough comes together to form a ball with nothing sticking to the side of the bowl.
Remove the dough ball from the bowl and onto a floured surface and knead by hand for about 6 to 8 minutes until the dough is sticky and elastic but does not stick to your fingers and come off.
Form the dough into a ball and place it into a large mixing bowl lightly oiled with olive oil. Cover bowl with aluminum foil. Let dough rise in a warm place until it has doubled in size; about 2 hours.
Transfer dough to a floured work surface and pat to flatten bubbles and form into a slightly rounded rectangle of dough about 5×10 inches and about ½ inch thick. Dust dough lightly with flour if needed. Cut dough into 8 equal pieces. Form each piece into a round bun shape.
Use your hands to gently pat and stretch the dough rounds into flat disc shapes about ½ an inch thick. Arrange buns about ½ an inch apart on a prepared baking sheet. Dust buns very lightly with flour. Drape a piece of plastic wrap over the baking sheet (do not seal tightly). Let buns rise until doubled, about 1 hour.
Preheat oven to 375 degrees F.
Bake the buns until lightly browned on top, 15 to 17 minutes. Buns will stick together slightly where they touch. Let cool completely, tear the buns apart, and slice in half crosswise to serve.