This one-pot dinner is all about efficiency: The chicken cooks with the pasta, then garlic, lemon juice, and red-pepper flakes add zest to a one-minute sauce.
In a large pot of boiling salted water, cook penne until al dente, adding chicken during last 2 minutes of cooking. Reserve 1/4 cup pasta water. Drain pasta and chicken; return to pot.
To pot, add tomato, tomato paste, garlic, oil, red-pepper flakes, and spinach; season with salt. Cook over medium, tossing, until spinach is barely wilted, about 1 minute. Remove pot from heat; stir in lemon juice and enough pasta water to create a thin sauce that coats pasta. Season with salt, if necessary, and serve immediately.