In a large nonstick skillet, cook the chorizo over medium-high heat until cooked through, breaking up the chorizo into 1/3- to 1/2-inch pieces with a wooden spoon, 12 to 14 minutes. Drain on paper towels and cool.
Heat the oil in the same skillet over medium heat. Add the garlic, carrots, celery and onions. Cook until the vegetables are tender but not at all brown, stirring often, 8 to 10 minutes. Add the salami and tomatoes. Bring the sauce to a simmer. Cover, reduce the heat to low and cook gently until the flavors blend, stirring occasionally, 15 to 20 minutes. Season with the 1/4 teaspoon salt and the pepper. Remove the skillet from the heat and add the arugula and basil. Stir until wilted.
Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, 8 to 10 minutes. Drain and reserve about 1 cup of the pasta water. Place the pasta into a serving dish.
Add the chorizo sauce to the pasta. Toss to combine, adding the reserved pasta water, if needed, to loosen the sauce.
Top with the Parmesan and serve.