How to Make It
Place dough on counter at room temperature; cover to prevent drying.
Preheat broiler to high.
Heat a 10-inch cast-iron skillet over medium-high heat. Coat pan with cooking spray. Add sausage; cook 3 minutes or until browned, stirring to crumble. Remove sausage from pan. Add kale, 2 tablespoons water, and sugar to pan; cover and cook 2 minutes or until kale wilts. Place kale on 2 layers of paper towels; squeeze out excess moisture. Wipe pan clean with paper towels.
Place nuts and 1 garlic clove in a mini food processor; pulse until finely chopped. Add kale; pulse until finely chopped. Add 2 tablespoons oil; process until almost pastelike (add 1 to 1 1/2 tablespoons water, if necessary). Add Parmigiano-Reggiano; pulse mixture just until combined.
Heat skillet over medium-high heat. Roll dough into a 10 1/2-inch circle. Add 1 tablespoon oil to pan; swirl to coat. Fit dough in pan, pressing slightly up sides of pan. Top evenly with pesto; sprinkle with sausage and mozzarella. Cook 2 minutes over medium-high heat or until browned on bottom. Place pan in oven; broil 2 minutes or until cheese melts. Cut into 8 wedges.
Combine tomatoes, remaining 1 tablespoon oil, and remaining garlic clove on a foil-lined jelly-roll pan; toss to coat. Broil 5 minutes or until tomatoes burst and are charred in spots, stirring occasionally. Place arugula in a large bowl. Top with tomato mixture (and juices), salt, and pepper; toss to combine.