Preheat oven to 500°. If using a pizza stone, place it on a rack in the oven.
Bring 2 qts. very hot water to a boil in a covered pot. Add kale; cook until tender to the bite, about 3 minutes. Drain, rinse with cold water, and squeeze out excess water.
Meanwhile, roll dough on a floured work surface into a 14-in. circle, letting it rest a few seconds between passes of the rolling pin so it will stretch. Lay dough on a floured baking sheet.
Sprinkle dough with manchego, chorizo, kale, and pepper. Slide pizza onto hot stone, if using, or bake on the baking sheet until browned and puffed, 5 to 10 minutes. Drizzle with oil.