1. Preheat oven to 450°.
2. Combine tomatoes, 1 1/2 teaspoons oil, 1/4 teaspoon rosemary, and 1/8 teaspoon black pepper in a small bowl, tossing to coat. Arrange tomatoes in a single layer on a jelly-roll pan; bake at 450° for 20 minutes, stirring once. Cool slightly.
3. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine 1 cup water, flour, 1 tablespoon olive oil, remaining 2 teaspoons rosemary, salt, and the remaining 1/4 teaspoon black pepper in a large bowl, stirring with a whisk until smooth. Let stand 30 minutes.
4. Place a 10-inch cast-iron skillet in oven.
5. Heat a large skillet over medium-high heat. Add 1 1/2 teaspoons olive oil; swirl to coat. Add onion; cover and cook for 8 minutes or until tender, stirring occasionally. Stir in 1/8 teaspoon red pepper and garlic. Cook, uncovered, 18 minutes or until onions are golden, stirring frequently. Remove from heat; stir in asparagus.
6. Carefully remove cast-iron skillet from oven. Add 1 tablespoon oil; swirl to coat. Stir batter once; pour batter into skillet. Bake at 450° for 20 minutes. Top evenly with onion mixture and tomatoes; sprinkle evenly with cheese. Bake an additional 12 minutes or until center of dough is set. Remove from oven; let stand 10 minutes. Cut into 4 wedges.
7. Combine remaining 1 tablespoon oil, juice, and vinegar in a medium bowl, stirring with a whisk. Add arugula; toss to coat. Arrange 1 cup arugula on each of 4 plates; sprinkle each serving with about 2 teaspoons nuts. Serve with 1 wedge farinata.