Combine the lemon juice, Dijon mustard, dried oregano, salt, and pepper in a bowl and whisk to combine. Slowly whisk in olive oil until emulsified. Add the mesclun greens to bowl with dressing and toss. Put greens on platter and set bowl aside with leftover dressing, to drizzle over other ingredients at end.
Slice tuna steaks on bias and arrange on top of greens. Scatter the olives and cherry or grape tomatoes over the salad. Add the reserved, cooked and cut Yukon gold potatoes and reserved green beans. Drain capers and scatter on top of greens. With kitchen shears, snip the parsley over the salad. Sprinkle shredded Parmesan over top of salad along with any leftover dressing.