In a large pot over medium-low heat, heat some oil. Add the carrots, onion, garlic and ginger. Season with salt. Saute for 15-20 minutes, stirring occasionally, or until the carrots are fork-tender. Move to a bowl and reserve.
In a large pot over medium heat, heat some oil. Add the shiitake mushrooms and a pinch of salt, then cook for 10-15 minutes, stirring occasionally, until any released liquid has evaporated and the mushrooms are darker. Add the soy sauce and toss the mushrooms until coated, and cook until the soy sauce looks slightly thickened, 1-2 minutes. Add the sesame oil, toss to coat, then move the mushrooms to the same bowl as the carrots.
In the same pot, mix the miso with a splash of the water until broken up, then stir in 4 ½ cups water and the barley. Place the pot over medium heat, and cook for 20-25 minutes, or until the barley is tender and chewy. Add all of the cooked vegetables and arugula, then mix until combined. Serve warm.