Prepare the barley: rinse barley and drain well. Place barley and vegetable broth in a small saucepan and bring to a boil over high heat. Keep an eye on this so it does not boil over. Once at a boil, lower the heat to maintain a simmer and cover. Allow to cook for 25-30 minutes or per package instructions. Add more water if the pan dries out, then drain any excess water after the barley is cooked. Cooked barley should have tripled in volume and become soft and chewy in texture. Fluff with a fork then set aside.
While barley is cooking, preheat the oven to 400℉ and line a baking sheet with parchment paper.
Transfer broccoli and carrots onto the parchment paper. Drizzle oil and season with oregano, the juice of half a lemon (about 2 tablespoons), and salt and pepper. Toss to coat and arrange seasoned vegetables in a single layer, spacing them out as much as possible.
Roast in the oven for 20-25 minutes, tossing halfway through at the 10 minute mark.
While waiting for the barley and vegetables to cook, prepare the tahini dressing: mix tahini, olive oil, maple syrup, black pepper, and the juice of the other half of the lemon (about 2 tablespoons) in a small bowl. Add water or more olive oil as needed to thin out the dressing.
Assemble your bowl: divide cooked barley into four bowls or containers. Top with roasted vegetables, arugula, and roasted sunflower seeds. Drizzle tahini dressing right before serving. Store your bowls separately from dressing in an airtight container in the fridge for up to 3-4 days.