Heat a 10-inch heavy (preferably cast-iron) skillet on medium-high until a drop of water bounces across it and immediately disappears. Add olive oil; tilt pan to coat bottom and sides. When the oil ripples across the bottom, it is ready for frying.
Add the fresh shiitake and cook until most of their water has been driven off and their flesh turns golden from the infused oil. When this occurs, add the soy sauce or tamari; stir until it bubbles and colors the mushrooms; add the sesame oil.
Turn the heat down to medium, scrape the mushrooms to the sides of the skillet, and add the fish, skin-side down. Cook for 4 minutes on one side; flip over, and cook until the skin side (now the upper side) feels firm. Remove fish from the skillet.
Add a bit more toasted sesame oil to the mushrooms in the pan and stir until they glisten.
Serve 3–4-inch slices of salmon with mushrooms on the side, a small green salad, bread, and potatoes or a grain.