Preheat your grill to medium heat. Brush the corn and scallions with 2 tablespoons of olive oil and sprinkle them with the salt and pepper. When the grill is hot, place the corn and scallions directly on the grates. Cook the scallions for 1 to 2 minutes, flipping them during cook time. Cook the corn until the kernels soften and turn golden brown, about 8 to 10 minutes, turning the corn as you go. [As a note, I do this while the water is coming to a bowl and the pasta is cooking.]
Once the corn is cool enough to touch, cut the kernels off the cob. Slice the scallions.
Heat a large skillet over medium heat and add the remaining 2 tablespoons of olive oil. Add the gnocchi and cook until the sides are golden brown and toast, flipping as you go, for about 3 to 4 minutes. Remove the gnocchi and add it to a bowl. Toss it in brown butter, then add the corn and scallions. Sprinkle the parmesan and basil leaves on top. Serve!