Heat the brown rice as the label directs; let cool slightly. Meanwhile, preheat the broiler and line a baking sheet with foil. Place the salmon skin-side down on the baking sheet and season with salt and pepper. Brush with 1 tablespoon ponzu sauce. Let sit at room temperature, 10 minutes.
Whisk the remaining 1/4 cup ponzu sauce with the sesame oil, pickled ginger and brine and wasabi paste in a large bowl. Set aside 3 tablespoons of the dressing for topping. Toss the rice with the remaining dressing; season with salt and pepper.
Broil the salmon until just cooked through and lightly charred on top, 5 to 8 minutes, depending on the thickness. Flake the salmon into large pieces, discarding the skin.
Divide the rice mixture among bowls. Top with the salmon, cucumbers and salad mix. Drizzle with the reserved 3 tablespoons dressing. Season with salt and pepper. Top with the seaweed and sesame seeds.