For pizza-oven-quality crust, be sure to get the skillet ripping hot.
Preheat oven to 500 degrees, with a 10-inch heatproof skillet (preferably cast iron) on rack in lower third. Once oven reaches temperature, let skillet continue to heat 15 minutes. Meanwhile, stir together cheeses, garlic, oil, 1/2 teaspoon salt, and a pinch of pepper.
On a lightly floured baking sheet, stretch dough to a 12-inch round. Remove skillet from oven and brush with an oiled paper towel. Carefully press dough into skillet, pushing up at edges to form a crust. Bake until bubbles form, 2 minutes. Remove from oven; top with cheese mixture, artichokes, and spinach, then drizzle with oil and season with salt and pepper. Bake pizza until bubbling and crisp on bottom, about 12 minutes more. Let cool slightly, then slice and serve.