Fill a medium pot with about 6 cups of water. Set the pot over high heat and bring the water to a boil.

Add 1 teaspoon of the kosher salt and the barley to the boiling water. Lower the heat so the water stays at a steady boil and cook 25 to 30 minutes, until the barley is tender but still pleasantly chewy.

While the barley cooks, put the rice vinegar and the remaining 1/4 teaspoon salt in a medium bowl and stir to dissolve the salt.

Add the sliced cucumber and shredded carrots to the bowl with the vinegar. Toss well so the vegetables are evenly coated.

Set the pickle bowl aside at room temperature. Toss the vegetables once or twice while you make the rest of the bowl.

If needed, slice the mushrooms. Slice the scallions thinly and keep the white parts and green parts in separate piles.

Put the gochujang, soy sauce, and sesame oil in a small bowl. Stir in 3 tablespoons of water until the sauce is smooth.

Heat the neutral oil in a large skillet over medium-high heat until it looks loose and shiny.

Add the mushrooms to the skillet in an even layer. Cook for 4 minutes with only a little stirring so they can brown.

Stir the mushrooms and keep cooking for 3 to 4 minutes, until they have released their liquid and the pan looks mostly dry again.

Add the scallion whites to the skillet. Cook for 1 minute, stirring, until they smell fragrant.

Pour half of the gochujang sauce into the skillet. Stir for 1 minute, until the mushrooms are glossy and lightly coated.