In the bowl of a stand mixer fitted with a dough hook attachment, add the flour, yeast, garlic powder and sugar.
Mix until combined. Heat the beer until the temperature reaches between 120°F and 125°F (double check your yeast package to confirm this is the temperature your yeast needs. Default to the temperature listed on the package).
Add the beer to the stand mixer and mix on medium speed. Once most of the flour has been moistened, slowly add the salt and oil while the mixer is still running. Turn speed to high and beat until dough is smooth and elastic, about 8 minutes. Transfer dough to a lightly oiled bowl, tightly wrap with plastic wrap. Allow to sit in a warm room until doubled in size, about 45 to 60 minutes.
Preheat the grill to 500°F.
Roll the dough out on a lightly floured surface to about 10-inches in diameter, add to a pizza peel covered with flour or cornmeal. Brush with olive oil.
Add to the grates of the grill, oiled side down. Close the lid and allow to cook until grill marks appear, use the pizza peel to remove the crust, transfer to a work station.
Cover with pesto sauce and sprinkle with parmesan cheese.
Return the pizza to the grill.
Shut the lid and allow to cook until the top of the crust is bubbly and starting to brown, about 6-8 minutes.
Remove from the grill, transfer to a serving platter, top with peaches, basil, burrata (pulled the burrata apart and distribute across the pizza), arugula, and balsamic glaze.
Slice and serve.