PREHEAT THE OVEN TO 400 DEGREES F: Set oven rack at center.
MAKE THE VINAIGRETTE: In a food process or blender, pulse together all of the ingredients except the olive oil until well mixed. With the machine running, add the olive oil slowly until it’s emulsified. You can also do this by hand with a whisk. Set aside.
PREP THE SALAD: In a very large bowl, toss together the romaine, cilantro, salt and pepper, and cup of Cheddar or goat cheese. Set the tortilla chips or croutons aside for later. Put the salad in the refrigerator while you make the fish.
PREP THE FISH: Put the flour, egg, and bread crumb mixtures into 3 separate shallow bowls. Mix the ½ teaspoon kosher salt, ¼ teaspoon fresh ground black pepper, and the 1/8 teaspoon ground cayenne in a small cup and sprinkle both sides of the tilapia fillets with the mixture, patting the spices into the flesh of the fish. Dip each fillet on both sides into first the flour mixture, then the egg mixture, and then the bread crumb mixture, shaking them off a bit after each and patting the crumbs into both sides of the fish. Place fillets at least 2 inches apart on a rimmed sheet pan lined with foil and sprayed with cooking spray. If one end of a fillet is a lot thinner at the end than it is in the middle, tuck that under to avoid over cooking.
BAKE THE FISH/MAKE SALAD: Bake for 10 minutes (no need to turn over) or until fish is firm, opaque, still juicy, and flakes when nudged with a fork. (approx. 140 degrees F on an instant read thermometer. The temperature will rise as the fish rests.) Meanwhile, divide the chilled romaine mixture between 4 bowls; add tomatoes, and avocado. Garnish with tortilla chips or croutons. Drizzle with vinaigrette.
PLATE/SERVE: Add a hot baked fish fillet to the center of each salad, drizzle with a little vinaigrette and garnish with the reserved grated Cheddar and cilantro sprigs. Serve hot or warm.