Pickled jalapenos do double duty here -- they're chopped for the pizza, and the brine is made into a dressing for the tangle of carrots, fennel, and arugula on top.
Preheat oven to 500 degrees, with an inverted rimmed baking sheet on top rack. Brush another baking sheet with 2 tablespoons oil; stretch dough into an 11-by-12-inch rectangle on it (if dough springs back, let stand a few minutes). Scatter sausage evenly over dough. Bake on inverted sheet 10 minutes. Sprinkle with cheese, jalapenos, salt, and pepper. Bake until crisp on bottom and cheese is bubbling, 5 to 8 minutes more. Let cool slightly.
Meanwhile, peel carrots and shave into short lengths with peeler. Thinly slice fennel. Toss carrots, fennel, and arugula with brine and remaining 2 tablespoons oil. Season with salt and pepper. Top pizza with a handful of salad just before serving, with remaining salad on the side.