Place the artichoke hearts in a bowl with 1/4 cup olive oil, 1 teaspoon salt and 1/2 teaspoon pepper and toss until the artichoke hearts are coated. Dump the artichoke hearts onto a sheet pan and spread out into 1 layer; roast in the oven for 20 minutes.
Meanwhile, make the vinaigrette. Place the minced shallot, lemon juice, mustard, 1 tablespoon vinegar, 1 teaspoon salt, and 1/2 teaspoon pepper in the bowl of a food processor fitted with the steel blade. Process for 5 seconds. Add the basil leaves and process into a green puree. With the processor running, slowly pour 1/2 cup olive oil into the bowl through the feed tube until the ingredients are finely pureed. Set aside.
Place the roasted artichoke hearts in a bowl and toss with enough vinaigrette to moisten. Add the capers, red peppers, red onion, parsley, 4 tablespoons vinegar and red pepper flakes, if using, and toss gently. Sprinkle generously with salt and pepper and let stand for 30 minutes for the flavors to blend. Serve at room temperature.