In a small bowl, combine 1/2 cup of the olive oil with the parsley, red onion, vinegar, basil, cilantro, garlic, crushed red pepper and 1/2 teaspoon of the thyme. Season this chimichurri sauce with salt and pepper.
Season the tuna steaks with salt, pepper and the remaining 1 teaspoon of thyme. Heat 2 large skillets. Add 1 tablespoon of the remaining olive oil to each skillet; when it shimmers, add 2 tuna steaks to each pan. Cook over high heat until browned but rare in the center, 1 to 2 minutes per side.
In a large bowl, toss the mesclun with half of the chimichurri sauce and mound it on plates. Thinly slice the tuna steaks and set on the plates. Pass the remaining chimichurri sauce at the table to dress the tuna.