Make the arugula pesto
Bring a large pot of water to a boil. In a food processor, combine the garlic, pine nuts, arugula, and ¼ cup (1 oz/30 g) of the Parmesan cheese. Pulse to chop. With the machine running, gradually add the oil. Season with ¼ teaspoon salt and several grindings of pepper.
Cook the pasta
Add 2 tablespoons salt and the pasta to the boiling water. Cook, stirring occasionally to prevent sticking, until al dente, according to the package directions. Drain, reserving about ½ cup (4 fl oz/125 ml) of the cooking water. Return the pasta to the pot.
Finish the pasta
While the pasta is cooking, in a large frying pan over medium-high heat, melt the butter. Add the chicken and cook, stirring occasionally, until heated through, 3–4 minutes. Add the chicken to the drained pasta along with the pesto, tomatoes, ricotta salata, and remaining ¼ cup Parmesan cheese. Toss, adding as much of the cooking water as needed to achieve a nice sauce consistency. Season with salt and pepper and serve.