Pour the water into a large bowl and add the yeast. Let sit for 2 minutes while the yeast activates. The mixture should become foamy and some of the yeast will sink in the water.
Add the eggs and oil. Whisk until smooth.
Add the gluten-free pizza dough mix and stir with a wooden spoon until well-combined.
Use a bowl scraper to divide the dough in half. Grease 2 medium bowls with olive oil. Place each dough half in a bowl and top with another drizzle of oil.
Toss the dough to evenly coat in oil, then cover the bowls with clean kitchen towels. Let rise for 20 minutes, until the dough has expanded, but not quite doubled in size.
Heat a grill pan over high heat.
Turn a dough portion onto a lightly floured pizza peel or other flat surface, like a cutting board or cookie sheet. Dust the dough with more flour and roll out into a 10-inch (25-cm) round, or whatever size will fit the pan you’re working with.
Carefully slide the dough onto the grill pan. Grill each side for 6-8 minutes, until dark grill marks form. Remove from the heat. Repeat with the remaining dough.