Preheat oven to 350 degrees F (175 degrees C).

Toss artichoke hearts with 1 tablespoon olive oil in a bowl to coat; spread onto a baking sheet. Sprinkle 1 teaspoon salt over the artichoke hearts.

Roast in preheated oven until they get a little brown around the edges, 15 to 20 minutes.

While the artichokes roast, vigorously whisk shallot, lime juice, vinegar, mustard, 1 teaspoon salt, and black pepper together in a bowl. Add basil; stir. Slowly stream 1/2 cup olive oil into the mixture while whisking.

Transfer artichoke hearts to a serving dish. Pour about 1/3 of the dressing over the artichokes and toss gently to coat. Add capers, bell pepper, and onion. Pour more dressing over the mixture until everything is coated, but not swimming in liquid. Top with balsamic glaze and shaved Parmesan cheese.