In a small bowl, whisk ginger, miso, soy sauce, sesame oil, 2 grated garlic cloves, 2 tablespoons vinegar, 3 tablespoons brown sugar, and a pinch of red pepper until smooth. Pat salmon dry; season both sides with salt. Spoon miso mixture over salmon and spread to coat. Refrigerate, uncovered, at least 30 minutes or up to 1 hour.

Meanwhile, in a medium bowl, toss cabbage with 1/2 teaspoon salt and remaining 1 teaspoon brown sugar; set aside until ready to use.

Fill a medium pot halfway with water; stir in 1 teaspoon salt. Bring to a boil over high heat and add farro. Reduce heat to medium and simmer, stirring occasionally, until farro is tender but slightly chewy, 20 to 25 minutes. Drain and return to pot. Cover and set aside until ready to use.

Place a rack in upper third of oven; turn on broiler. Line a large sheet tray with foil, then place a wire rack on tray. Arrange scallions on rack and drizzle with 1 tablespoon neutral oil; season with salt. Wipe off a majority of marinade from salmon and arrange on rack next to scallions.

Broil salmon and scallions, watching closely, until salmon is cooked through and flesh easily flakes with a fork and scallions are charred, 10 to 12 minutes. Let cool slightly.

Transfer scallions to a cutting board. Finely chop scallions and transfer to a medium bowl. Add cilantro and remaining 1 grated garlic clove, 4 tablespoons neutral oil, 2 tablespoons vinegar, and pinch of red pepper and whisk to combine; season with salt. Add 2 tablespoons scallion dressing to pot with farro and toss until combined.

Add cucumber and carrots to bowl with cabbage and toss with remaining scallion dressing. Divide farro among bowls or plates. Top with salmon and cabbage salad.