In a small saucepan, heat the milk until just simmering, then turn off the heat. Allow to cool on the counter or in the refrigerator until around 105°F, no warmer than 110°F.

Combine the warm milk, yeast, and sugar in the bowl of a stand mixer fitted with the dough hook and allow to rest for 10 minutes. The mixture should appear foamy.

Add the egg, flour, and salt, then turn the mixer on low speed to combine. Mix for 5 minutes on low speed.

Turn off the mixer and scrape the bowl and dough hook well. Add the olive oil, and continue to mix for another 10-15 minutes, until the dough is smooth and stretchy but not overly sticky. The dough should stretch, not break when the dough hook is lifted.

Tuck the dough into a ball and place it in a lightly greased bowl. Cover and allow to rise until doubled, 60-90 minutes at a warm room temperature.

Gently punch down the dough, then divide into 4 even pieces. Tuck each piece into a ball, and allow to rest, covered, for 5 minutes.

On a lightly floured surface, roll each piece of dough into a 10-inch circle.

Place one dough circle onto a baking sheet lined with parchment paper. Spread 2 tablespoons of pesto onto the dough, leaving a ½ inch border at the edge. Place the next dough circle on, and repeat with 2 more tablespoons of pesto. Place the third dough circle on top, and add the last 2 tablespoons of pesto.

Place the final dough circle over the last pesto layer, and use the edges of your hands to gently press all the way around the edges of the dough to lightly seal the layers together. Be careful not to trap large air bubbles between the layers.