Place yeast into the bowl of a large stand mixer; whisk in 1/2 cup flour and warm water until smooth. Set aside until mixture is foamy, 10 to 15 minutes.

Whisk 1 egg, melted butter, sugar, and salt thoroughly into yeast mixture. Add remaining 3 cups of flour.

Knead dough in the bowl of the stand mixer fitted with the dough hook on low speed until soft and sticky, 5 to 6 minutes. Scrape down the sides of the bowl if needed. Poke and prod dough with a silicone spatula; if large amounts of dough stick to the spatula, add a little more flour.

Transfer dough onto a floured work surface; dough will be sticky and elastic but should not stick to your fingers. Gently form dough into a smooth, round shape, tucking loose ends underneath.

Wipe out the stand mixer bowl; drizzle in olive oil. Place dough in the oiled bowl; flip several times until the surface is thinly coated with oil. Cover the bowl with aluminum foil; let dough rise in a warm place until doubled in volume, about 2 hours.

Line a baking sheet with a silicone mat or parchment paper.

Transfer dough to a floured work surface; pat to flatten any bubbles and form into a slightly rounded, 5x10-inch rectangle, about 1/2 inch thick.

Dust dough lightly with flour if needed; cut into eight equal pieces. Form each piece into a round shape; gently tuck ends underneath as before.

Gently pat and stretch dough rounds into flat, 1/2-inch-thick bun discs using your hands. Arrange buns about 1/2 inch apart on the 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 in volume, about 1 hour.

Preheat the oven to 375 degrees F

Beat remaining 1 egg and milk together in a small bowl, using a fork, until mixture is thoroughly combined. Very gently and lightly brush tops of buns with egg wash without deflating risen dough. Sprinkle buns with sesame seeds.

Bake in the preheated oven until lightly browned on tops, 15 to 17 minutes. Buns will stick together slightly where they touch. Cool completely. Tear buns apart. Slice in half crosswise to serve.