In a large bowl or a bowl of a stand mixer, combine the warm water, 1 tablespoon sugar, and yeast. Whisk to dissolve the yeast. Leave for 5-10 minutes until foamy.
Whisk in the remaining sugar, egg, egg yolks, honey, oil, and salt. Slowly add the flour. Knead the dough until smooth with the hook or by hand for 8-10 minutes.
Place the dough in a greased bowl. Cover the bowl with plastic wrap and let rise in a warm, draft-free place until almost doubled in size, about two hours.
While challah is rising, make pesto. Pulse basil in a food processor with olive oil until a fine paste. Add the remaining ingredients and pulse until smooth. Adjust salt as desired. Set aside.
Divide the dough into three pieces. Roll each piece of dough into a long rope. If the ropes shrink as you try to roll them, let them rest for 5 minutes to relax the gluten, and then try again.
Flatten each rope and fill with pesto. Seal up each strand, pinch the three strands at one end, and carefully braid.
Line a baking sheet with parchment paper. Carefully transfer challah to a parchment-lined baking sheet.
Cover loosely with plastic, then let it stand until soft and fluffy, about 30 minutes.
Preheat oven to 350°F.
Whisk the egg yolk with water and brush over challah. Sprinkle the sesame seeds on top.
Bake for 30-35 minutes until golden brown.