On a large rimmed baking sheet, drizzle 1 tablespoon of the olive oil. Add the pizza dough and turn to coat. Cover with a clean kitchen towel and let sit at room temperature for 1 hour.
Meanwhile, set a fine mesh sieve over a small bowl. Drain the marinated artichoke hearts, reserving the oil. In a small bowl, toss the pecorino with the crushed red pepper and lemon zest, reserving the zested lemon half for serving. In a separate small bowl, whisk the grated garlic with the remaining 2 tablespoons of olive oil.
Preheat the oven to 500°F. Using your hands, stretch the dough to fill the entire baking sheet (the dough should be thin). If the dough springs back, let it rest for a few minutes before stretching again. Brush the top of the dough with the garlic oil.
Scatter the mozzarella over the pizza, leaving a ½-inch border. Tear the marinated artichokes into bite size pieces and scatter them over the mozzarella. Sprinkle the pecorino mixture over the top.
Bake the pizza until the dough is golden and puffed and the cheese is melted and beginning to brown in spots, 10 to 15 minutes. Remove from the oven and let cool slightly.
When you’re ready to serve, scatter the arugula and herbs over the top of the pizza. Drizzle some of the reserved artichoke oil over the greens and squeeze the reserved lemon half over the top, then sprinkle with flaky sea salt. Cut into squares and serve right away.