Preheat your oven to 500°F. While it heats, prepare your mise en place: dice the red onion into thin slices, halve the tomatoes, and drain the artichoke hearts well (excess moisture will make the flatbread soggy). Roughly chop the parsley and have the arugula ready for garnishing at the end.
Place your flatbreads on a baking sheet. Brush both flatbreads evenly with 2 tablespoons of olive oil on one side—this creates a crispy, flavorful base. Spread the spinach pesto generously and evenly across each flatbread, leaving a small border. I like to use the back of a spoon to spread pesto smoothly so it distributes flavoring throughout every bite.
Tear or slice the mozzarella into even pieces and distribute it across both flatbreads as your first cheese layer—this will help the other toppings adhere and melt together. Next, scatter the drained artichoke hearts, black olives, and diced red onions evenly over the mozzarella. Distribute the tomato halves cut-side up across the surface, then crumble the feta cheese over everything. Sprinkle with oregano, red pepper flakes, and a pinch of salt to taste.
Transfer the baking sheet to the preheated 500°F oven and bake for 5-10 minutes, depending on your oven. Watch for the mozzarella to melt and the flatbread edges to turn golden and slightly crispy. The exact time varies, so check around the 5-minute mark. I remove mine when the cheese is bubbly and the crust starts to brown at the edges—this takes about 7-8 minutes in my oven.
Remove the flatbread pizzas from the oven and let them cool for just 1-2 minutes so they're easier to handle. Scatter the fresh parsley and arugula over the top while the pizzas are still warm—the heat will slightly wilt the arugula and release its peppery flavor. Drizzle with the remaining 1 tablespoon of olive oil for a final touch of richness, then slice and serve immediately.