Chop fresh broccoli into small florets. If using frozen, let them stand out for a few minutes, then give a quick coarse chop. Measure and set aside.
Set a large colander in the sink with a glass mug or liquid measuring cup in the center of the colander (as a reminder to reserve some pasta water before draining it!). Bring a large pot (5.5-quart or bigger) of water to boil. Once boiling, generously salt water to season (I add 1 tablespoon salt.) Return to a boil and add penne. Cook according to package directions, subtracting one minute. Two minutes before draining, add in chopped broccoli. Right before draining, remove 1 cup pasta water and then drain. Do not rinse.
Meanwhile, thinly slice shallots. Add 1 tablespoon butter to a large pot or sauté pan (big enough to hold all the pasta and vegetables) over medium heat. Once melted, add shallots and a pinch of salt and pepper. Cook, stirring frequently, about 3 minutes. Add garlic and stir until fragrant, 30 seconds. Add 1/4 cup reserved pasta water (just dip a measuring cup into the pot if the pasta is still boiling). Stir. Add in can of creamed corn and mix. Simmer 3–5 minutes, stirring frequently. Use a spatula to scrape all the mixture to a blender. Blend mixture until very smooth (1–2 minutes on high). See note 4.
While mixture is blending, return the pot/pan to medium-high heat. Add remaining 3 tablespoons butter and melt. Once melted, add frozen corn, grated carrots, and red pepper strips and cook, 3–4 minutes or until crisp tender. Scrape all the blended corn mixture into pan and mix. Reduce heat to medium and simmer to combine flavors until pasta is done cooking.
Add drained pasta and broccoli and 1/4 cup of the reserved pasta water and toss vigorously to coat. Toss and cook for 1–2 minutes, then add a little more pasta water if the mixture seems too thick. While tossing, gradually add Parmesan cheese, 2 tablespoons at a time. Continue to toss until cheese is melted and sauce is thick and coats the pasta nicely. Taste and add additional salt and pepper to taste (I typically add 1/2 up to 1 teaspoon pepper and another pinch of salt). Mix in basil.
Serve plates of pasta, garnished with additional basil and Parmesan if desired and a sprinkle of red pepper flakes if using. Enjoy immediately!
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Notes
Note 1: Grate large carrots on the large holes of a cheese grater. Measure and add.Note 2: Grate a block of Parmesan on the small holes of a grater. Alternatively, use finely grated Parmesan cheese. Parmesan from the can will be too salty and won’t melt into the sauce nicely.Note 3: The basil adds so much to this recipe! I wouldn’t recommend leaving it out, but if you aren’t a fan, try flat-leaf Italian parsley instead.Note 4: When blending, make sure to secure the lid on tightly and select the “soup” or “hot” cycle. Increase speed slowly and watch it carefully to avoid sauce exploding. Remove the lid carefully as it will likely release a burst of steam. If your blender doesn’t have a hot/soup setting, you can replace the blender lid with a folded towel and hold it in place with your hands. This will help reduce steam pressure.Storage: Store leftovers in an airtight container in the fridge for up to 3-4 days. Reheat in the microwave or on the stovetop with a splash of water or broth to keep the sauce creamy.