Fill a large skillet with enough water to reach a depth of 2 inches. Ask your Grown-Up Helper (GH) to bring it to a boil over medium-high heat. Add the Broccolini florets and cook for 2 minutes. Ask your GH to drain in a colander. Wipe out the skillet.
Ask your GH to bring a large pot of water to a boil and salt it. Add the spaghetti and cook until al dente. Ask your GH to drain.
While the pasta is working, ask your GH to heat the skillet over medium heat. Add the butter to melt, then add the tomatoes and gently mash with a potato masher or a wooden spoon. Season with salt, stir in the basil, lower the heat and simmer for 4 minutes. Add the Broccolini and cook for 3 minutes more. Add the pasta and some cheese and toss. Pass more cheese around the table.