In a large mixing bowl, combine the 175 grams warm water, 10 grams active dry yeast, and 15 grams sugar. Let sit for 5 minutes, or until foamy.
Add the 1 beaten egg, 100 grams sourdough starter, and 25 grams olive oil. Stir until combined.
Add the 140 grams freshly milled khorasan flour, 240 grams all purpose flour, and 1 teaspoon sea salt. Mix until a shaggy dough forms. Cover and let rest for 30 minutes.
Turn the dough onto a lightly floured surface and knead for about 5 minutes, or until smooth and elastic.
Place the dough in a lightly oiled bowl. Cover and let rise for about 45 minutes, or until doubled in size.
While the dough rises, make the balsamic glaze. In a small saucepan, whisk together the ½ cup balsamic vinegar, 2 tablespoons honey, ¼ teaspoon red pepper flakes, and ¼ teaspoon salt.
Bring to a simmer over medium/low heat and cook for about 10 minutes, or until reduced and thickened. Set aside to cool, and keep in mind it thickens more as it cools. If you cook this too vigorously your glaze will thicken too much, keep an eye on it!
Preheat your grill to 500°F (260°C) and allow it to heat for at least 15 minutes.
Generously flour a pizza peel or cutting board. Place the dough on the floured surface and lightly oil your hands and the top of the dough.
Stretch the dough into a 12 to 14 inch oval about ½ inch thick.
Lightly brush or spray the top with olive oil.
Carefully flip the dough directly onto the hot grill grates.
Grill for 4 to 5 minutes. Flip and cook for another 3 minutes.
Spread the ½ cup pesto evenly over the flatbread and top with the 2 cups mozzarella cheese.
Close the grill lid and cook for about 1 minute, or until the cheese begins to melt.
Remove from the grill and drizzle generously with the balsamic glaze.
Slice and serve warm.