Prepare the Teriyaki Sauce: Combine 1 Tbsp sake, 1 Tbsp mirin, 2 Tbsp soy sauce, and 1 Tbsp sugar in a microwave-safe bowl, then mix until the sugar is well incorporated.
Dissolve the Sugar: Microwave the sauce mixture for 30 seconds if needed to help the sugar dissolve completely.
Season the Salmon: Rinse the 2 skin-on salmon fillets and pat them dry. Season one side with some of the ¼ tsp kosher salt and ⅛ tsp black pepper.
Season the Second Side: Turn the salmon over and season the other side with the remaining salt and black pepper.
Coat the First Side: Sprinkle half of the 1 Tbsp flour over one side of the salmon and spread it into an even coating.
Finish the Flour Coating: Turn the fillets over, coat the second side with the remaining flour, gently press the flour onto the fish, then remove any excess.
Heat the Pan: Place a frying pan over medium heat and allow it to become hot before adding ½ Tbsp neutral oil and 1 Tbsp unsalted butter, taking care not to let the butter burn.
Sear the Salmon Skin: Press the skin side of each fillet against the hot pan surface for 15 seconds, one fillet at a time, then place each fillet skin side down in the pan. Keep the heat controlled so the butter does not burn.
Brown the Salmon: Cook the salmon for about 3 minutes, until the bottom is golden brown, then carefully flip the fillets.
Steam the Salmon: Add 1 Tbsp sake to the pan, cover with a lid, and reduce the heat to low. Steam for about 3–5 minutes, depending on the thickness of the fillets. Adjust the Steaming Liquid: If the sake evaporates before the salmon is fully cooked, add another tablespoon of sake and continue steaming.
Check the Salmon: Cook until the thickest part of the fillet registers 125–130°F (52–54°C) on an instant-read thermometer, then transfer the salmon to a tray or plate. Fillets around ¾–1 inch (2–2.5 cm) thick took roughly 4 minutes in the source recipe. Reduce the Teriyaki Sauce: Add the prepared teriyaki sauce mixture to the pan and increase the heat slightly. Once the sauce begins to boil, return the salmon to the pan. Glaze the Salmon: Spoon the thickening sauce over the salmon, allowing it to coat the fillets. When the sauce has thickened, turn off the heat. Finish the Dish: Transfer the salmon to a serving plate and drizzle the remaining teriyaki sauce over the top. Serve immediately.