Make the dressing by combining mayonnaise, dried oregano (or fresh), 1/4 teaspoon salt and pepper, minced garlic, and lemon juice in a small bowl; mix until smooth and set aside.
Pat the tuna steaks dry and rub both sides with 2 tablespoons olive oil, then season with salt and pepper to taste.
Preheat a grill pan or skillet over medium-high heat and sear the tuna 2–3 minutes per side, until charred outside but still rare inside.
Transfer the tuna to a plate and let rest 5 minutes, then slice into strips.
For tortilla wraps: place a tortilla flat and spread about 1 tablespoon of the dressing down the center.
Layer carrot sticks, sliced avocado, cucumber, red onion, jalapeño, and optional spinach, green onions, and Parmesan along the tortilla center.
Top the vegetables with sliced tuna, fold in the sides, then roll to close the wrap.
Optional: return the wrapped tortilla to the grill pan and cook 2–3 minutes per side to warm and crisp the outside.
For lettuce wraps: open a large Boston lettuce leaf, spoon a teaspoon of dressing into the base, add the fillings and tuna, then drizzle another 1/2 teaspoon of dressing and fold to serve.
Serve immediately and enjoy.