In a food processor, combine the basil, parsley, garlic, lemon zest, a big pinch of salt and a few grinds of pepper. Pulse until everything is thoroughly chopped and combined. Add the butter and pulse until thoroughly mixed. Scrape the butter mixture out of the food processor onto a piece of plastic wrap or parchment paper and form into a log. Store in the refrigerator until ready to use.
Preheat a grill or a grill pan for medium-high heat. Lightly oil the grill grates. Grill the corn, rotating it as it cooks, until lightly charred in places, 3 to 4 minutes per side.
Remove the corn from the grill, smear each ear with a generous amount of the herb butter and serve.
You may alternatively cook the corn by bringing a large pot of water to a boil. Season the water with 2 tablespoons kosher salt, then boil the ears of corn until firm but tender, 3 to 4 minutes. Carefully remove the corn with tongs, drain on a paper towel-lined plate and serve with the herb butter.