To prepare the melted leeks, place a large skillet over medium-high heat with 1 tablespoon olive oil. Add the mushrooms to the pan and cook, stirring only occasionally, until they're golden brown and beginning to shrink a bit, 6 to 7 minutes. Remove the mushrooms from the skillet and reserve.
Return the skillet to medium heat with 1 tablespoon olive oil and the butter. Add the leeks to the pan, season them with salt and pepper, then cook, stirring frequently (don't let them get brown), until they've softened, about 10 minutes. Add the white wine and dried oregano to the skillet along with the reserved mushrooms and continue cooking, stirring occasionally, until most of the wine has evaporated, about 5 minutes more. Season as needed with salt and pepper; reserve warm.
While the leeks are cooking, prepare the gremolata. In a small mixing bowl, combine all of the ingredients and reserve.
When the leeks are just about ready, place a second medium skillet over medium-high heat with the olive oil. Season the fish liberally with salt and pepper, and sear it flesh-side down first, turning the fillet only once, until cooked to your liking, about 5 minutes per side for medium (depending on the thickness).
Serve the melted leeks topped with a piece of fish per plate and a garnish of the gremolata.
(NOTE: Don't salt the mushrooms while they're browning. The salt will pull the moisture out of them and they'll end up stewing instead of getting a nice golden-brown sear.)