Whisk the yeast and 1 teaspoon honey into 2/3 cup warm water (110 to 116 degrees F) in the bowl of an electric stand mixer, and let it stand for a few minutes until foamy. Combine the yeast mixture with remaining honey, 1/3 cup olive oil, and eggs. Add the salt and flour, switch to a dough hook, and run at low speed for 8 minutes. Transfer the dough to an olive-oil coated bowl, cover with plastic wrap, and set aside for 1 hour to rise.

After your dough has risen, turn it out onto a floured counter and divide it in half. Roll the first half of the dough into a really wide and imperfect rectangle. Spread half of the pesto filling evenly over the dough, stopping short of the edge. Roll the dough into a long, tight log. Then gently stretch the log as wide as feels comfortable (about two to three feet) and divide it in half. Repeat with remaining dough and pesto filling.

Arrange two ropes in each direction, perpendicular to each other, like you are doing a kindergarten paper -weaving project. Weave them so that one side is over, and the other is under where they meet. Take the four legs that come from underneath the center and move the leg to their right. Take the legs that were on the right and, again, jump each over the leg before, this time to the left. Continue this process until almost all the dough has been woven in. Tuck the odd ends under the dough to form a round.

Transfer the dough to a baking sheet lined with parchment paper. Beat egg in a small mixing bowl, and brush over challah. Let challah rise for another hour.

Preheat your oven to 375°F.

Before baking, brush loaf one more time with egg wash and sprinkle with sea salt. Bake in middle of oven for 35 to 40 minutes. If about halfway through the baking process the loaf looks too dark, cover it with tin foil, and continue baking. The center of the loaf should be 195 degrees F.

Cool loaf on a rack before serving.