Whisk 1/4 cup olive oil, 2 tablespoons brown sugar, 4 minced garlic cloves, 1/2 teaspoon salt, and 1/2 teaspoon pepper in a bowl to make the marinade.
Place the skirt steak in a baking dish or zip-top bag and pour the marinade over it. Refrigerate for at least 30 minutes or up to overnight.
Remove the steak from the refrigerator about 30 minutes before cooking to come toward room temperature.
Preheat a grill, broiler, or heavy skillet. If broiling, move the oven rack close to the broiler and preheat the broiler.
Grill, broil, or pan-sear the steak until desired doneness, about 5 minutes per side under a broiler for medium (timing will vary by method and thickness).
Transfer the steak to a cutting board and let it rest for 10 minutes before slicing thinly against the grain.
Make the avocado pesto: toast the pine nuts in a dry skillet over low heat, stirring, until golden and fragrant, about 5 minutes.
In a food processor combine 4 cups fresh basil, 1 avocado, 1/2 cup grated Asiago, toasted pine nuts, and 3 garlic cloves. Pulse until coarsely chopped.
With the processor running, stream in 1/2 cup olive oil and add up to 3/4 cup total as needed for a smooth but slightly chunky pesto. Season with 1/4 teaspoon salt, 1/4 teaspoon pepper, and 1/4 teaspoon crushed red pepper flakes; blend and taste, adjusting seasoning if needed.
Make the grilled corn relish: combine corn kernels (from 3 ears), 1/2 diced red bell pepper, 1 diced shallot, 1/4 cup chopped cilantro, and juice of 1 lime in a bowl.
Season the relish with 1/2 teaspoon salt and 1/2 teaspoon pepper; stir to combine and chill briefly if desired.
To serve, slice the rested steak thinly against the grain and top with spoonfuls of avocado pesto and grilled corn relish.