Combine 2 cups herbs, 1/4 cup scallions, ginger, and garlic in a food processor. Pulse to chop ingredients. Add 4 Tbsp. each of oil and vinegar, and 1/4 cup water. Process until a coarse purée forms. Transfer herb sauce to a bowl and stir in sesame seeds. Season to taste with salt and pepper. Herb sauce can also be made ahead. Cover and chill.
Heat 1 Tbsp. oil in large heavy skillet over medium-high heat until oil just begins to smoke. Add remaining 1/4 cup scallions and wheat berries, farro or brown rice. Stir-fry until grains are golden brown and slightly crunchy, about 3 minutes. Season with salt and pepper. Divide grain of choice among bowls.
Return skillet to medium-high heat and add remaining 2 Tbsp. oil. Add vegetables, season with salt and pepper, and stir-fry until brightly colored and crisp-tender, about 4 minutes. Add remaining 1 cup herbs and toss to combine. Divide vegetables over grain bowls. Drizzle with herb sauce.