To make the dough, in the bowl of a stand mixer, combine water and yeast. Let sit for about 10 minutes. Mix in bread flour, sugar, and kosher salt. As the mixer runs, add in olive oil. Beat until the dough forms into a solid ball. Transfer to a large greased bowl and cover in a warm place. Let double in size for about an hour. Once the dough has doubled in size, turn out onto a lightly floured work surface and divide into two equal pieces. Cover each piece of dough and let rest for 10 minutes.
As the dough rests, mix all of the pistachio basil pesto ingredients together in a food processor. Pulse until combined then set aside.
Preheat oven to 475F. Place each piece of dough onto a separate prepared baking sheet. Working with your hands, gently spread out dough working from the center out.
Top flatbread starting with the pistachio basil pesto. Sprinkle on dairy-free cheese followed by other toppings. Bake for 15 minutes and enjoy!