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1
In a blender or food processor, combine garlic, Parmigiano, basil, and oil and blend until very smooth, scraping down sides as needed and adding a tablespoon or two of water if needed to get things moving. Add lemon zest and season with salt and pepper. Set pesto aside (you should have about 1/2 cup).
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2
Bring a large pot of water to a boil for the pasta and season it generously with salt. In a medium Dutch oven or large cast-iron skillet, heat butter and oil over medium-high heat. When butter is foaming, add garlic, shallots, and reserved basil stems and cook until mixture is fragrant, about 2 minutes. Add wine and bring to a simmer. Add clams, toss to coat, cover, and cook, checking and stirring every few minutes, until clams begin to open, 3 to 5 minutes. As clams open, transfer them with a slotted spoon to a medium bowl, but try to keep pot covered as much as possible. Fish out basil stems and keep pan sauce warm over very low heat. (If you want a shell-free dining experience, gently liberate clams from their shells and return them to bowl with a splash or two of pasta water over them so they don't dry out.)
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3
Cook pasta until 1 or 2 minutes shy of al dente. Scoop 1 cup of pasta water into pan with clam sauce. Drain pasta and add to sauce along with half of the pesto. Add corn and bring mixture to a simmer. Cook, stirring, until liquid starts to thicken and pasta is al dente, about 2 minutes. Add clams and any accumulated liquid from bowl. Cook, stirring and tossing, until clams are warmed through, 2 minutes. Serve pasta and clams topped with more pesto.
