Dredge in flour being sure to coat both sides but more importantly, shaking off any excess flour.
Heat saute pan, add olive oil and let that get to medium high, not smoking.
Add the tuna to the pan and sear for about 1 1/2 minutes. Flip the tuna steaks over and cook on the other side for another minute. Repeat until all sides are seared.
Remove tuna from the pan to a plate while you make the sauce. Lightly cover with aluminum foil to keep semi warm.
Add the frying pan back to the burner over medium heat. Add the remaining 2 tablespoons of olive oil and heat. Add the garlic. Once the garlic is fragrant, add chopped anchovy fillets and oyster mushrooms.
When those ingredients start to sizzle, add the white wine, lemon juice, and broth. Add the thyme sprigs and taste. Season with salt if necessary. Bring the sauce to a low boil and cook until it is reduced by half.
Carefully slice tuna portions. Plate the slices into fallen domino like formation. Add butter to the sauce and stir to combine the flavors.
Remove the pan from the heat, add the parsley and blend.
Pour the hot sauce over the tuna.
Notes
Better than Bouillon makes a mushroom stock. Using chicken broth will not harm the flavor at all. You can substitute shiitake or cremini mushrooms for the oyster.