For the farro: Combine the farro with the water in a large saucepan, cover and bring to a boil over medium-high heat. Reduce the heat and simmer, covered, until al dente, 20 to 30 minutes. Drain off any excess water and transfer to a medium bowl. Season with salt.
For the salmon: Heat the olive oil in a large ovenproof skillet over high heat. Pat the salmon dry and season with 1 teaspoon salt and 1/2 teaspoon black pepper. Add the fillets to the skillet, skin-side down, and cook, undisturbed, for 2 minutes. Transfer the skillet to the oven and bake the salmon, still skin-side down, until the skin is crisp and golden but the flesh is still just translucent in the center, 3 to 5 minutes.
Meanwhile, make the salsa verde: Use both the blade edge and a flat side of a chef's knife to mince and mash the garlic, anchovies and salt to a paste. Transfer the paste to a food processor. Add the mustard, herbs and red pepper flakes, and pulse while slowly drizzling in both oils. The consistency should be chunky. Season with salt and black pepper and set aside.
For the assembly: Divide the arugula among 4 plates; top each with 1 salmon fillet, skin-side up. Sprinkle about 1/4 cup of the farro over each serving and drizzle generously with the salsa verde (about 2 tablespoons per plate). Halve the lemon, remove the seeds, and squeeze the juice over the salsa verde.