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In a large skillet over medium-high heat, brown the chicken with the mushrooms, onion and garlic in oil. Add the chicken broth and bring to a boil. Let simmer for 10 minutes. Add the tomatoes and cook until they start to fall apart. Add the arugula and cook for 1 minute. At this point, about one-third of the cooking liquid should be left in the skillet. Add more broth if required.
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While the chicken and vegetables are cooking, in a large saucepan of boiling salted water, cook the pasta until it is al dente. Drain.
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Transfer the pasta into the skillet. Add the Parmesan and combine well. Season with salt and pepper.
NOTE
Spätzle is a type of egg noodle sold at select supermarkets and European grocery stores. German in origin, it is found across much of Europe, including France, Switzerland, Austria, Hungary and the northern tip of Italy