Combine the warm water, yeast, and sugar in the bowl of a stand mixer fitted with the hook attachment. Let bloom for 10 minutes. This can also be done in a large bowl and kneaded by hand.
Add the egg, sourdough discard, olive oil, honey, and salt. Whisk to combine. Add the flour, starting at 2 ¾ cups, and mix until a shaggy dough then knead until a soft and pliable dough. If the dough remains very sticky, add ¼ cup more flour and knead 1-2 more minutes.
Cover and let rise in a warm environment for 1 to 1 ½ hours, until doubled in size.
Preheat the grill to 400F to 450F.
Divide the dough into two equal pieces for easier cooking.
Lightly flour a sheet of parchment paper on a large cutting board. Add the dough to gently stretch and press out to ¼” thick.
Generously drizzle with olive oil, then flip onto the grill with the help of the parchment paper. Cook for 2-3 minutes on each side. Use a large spatula and tongs to help flip without breaking.
Layer the grilled flatbread with a generous slather of basil pesto, then the burrata, marinated artichokes, sauteed asparagus, sliced radishes, heirloom tomatoes (seasoned with salt and black pepper), halved snap peas, and fresh dill. Enjoy warm.