Set a 6-quart / 5.7 L pot of water over high heat, add 1 tbsp kosher salt and bring to a boil. While it heats, cut the chicken into 2 cm / 3/4-inch cubes and toss with 1/2 tsp salt and 1/4 tsp pepper. Halve the tomatoes, mince the garlic and grate the Parmesan.
Add the cottage cheese, pesto, Parmesan, garlic, lemon juice and 1/4 tsp pepper to a blender. Blend on high for 60 seconds, scraping the jar once, until the sauce pours smoothly with no visible curds. Taste before adding any salt.
Add the penne or rotini to the boiling water and cook for 9 to 10 minutes, until al dente. Scoop out 120 ml / 1/2 cup of pasta water, then drain.
While the pasta cooks, heat the olive oil in a 30 cm / 12-inch skillet over medium-high heat until it shimmers. Add the chicken in a single layer and cook for 6 to 8 minutes, stirring once or twice, until browned and 165 F / 74 C in the thickest piece.
Lower the skillet heat, add the tomatoes and spinach, and cook for 1 to 2 minutes until the spinach wilts. Return the drained pasta to the empty pot, off the heat. Add the chicken mixture and the sauce, and toss for 1 minute, stirring in pasta water 2 tablespoons at a time until the sauce coats the pasta like loose cream. Serve right away with extra Parmesan.