Combine artichoke hearts, mayonnaise, Parmesan cheese, lemon zest, lemon juice, red pepper flakes, salt, and black pepper in a food processor or blender. Blend until smooth. Cover and refrigerate until ready to use.
Heat olive oil in a large pot or Dutch oven over medium-high heat. Stir tri-tip steak slices into the hot oil and cook, stirring constantly until they begin to brown, about 3 minutes. Sprinkle with Italian seasoning, salt, and black pepper. Add onion, yellow pepper, orange pepper, sweet hot peppers, and garlic basil spread; cook and stir until onion begins to soften, about 5 minutes. Stir in mushrooms, capers, Marsala wine, and anchovy fillet and cook until mushrooms are browned and wine has evaporated, about 5 minutes.
Place provolone cheese and Gorgonzola cheese on top of the hot beef mixture and gently fold until cheese is melted, about 2 minutes. Remove beef-cheese mixture from heat.
Spread each hoagie bun with about 1/4 cup artichoke aioli. Spoon beef-cheese mixture into buns and garnish with basil.
