Preheat the oven to 400°F (200°C).
On a large rimmed baking sheet, toss 1 cup diced carrots and 2 cups broccoli florets with 1 teaspoon sesame oil, 1/4 teaspoon kosher salt, and 1/4 teaspoon ground black pepper. Spread in a single layer and roast 20–25 minutes, until the carrots are soft and the vegetables begin to char at the edges. Remove from oven and set aside.
Meanwhile, drain and rinse the 14-ounce can of garbanzo beans (chickpeas) and pat them dry with a paper towel. On a second baking sheet, toss the chickpeas with the remaining 1 teaspoon sesame oil, remaining 1/4 teaspoon kosher salt, and remaining 1/4 teaspoon ground black pepper. Spread in a single layer and roast 10–15 minutes, just until they begin to brown and roast but are still somewhat soft. Remove from oven and let cool slightly.
While the vegetables and chickpeas roast, make the peanut sauce: in a food processor combine 2 tablespoons toasted sesame oil, 1/4 cup low-sodium soy sauce, 1/4 cup honey, 2 minced garlic cloves, 1 teaspoon crushed red pepper, and 1/4 cup creamy peanut butter. Blend until smooth.
Transfer the blended sauce to a bowl and gently stir in the 1/4 cup roasted unsalted peanuts. Reserve about 1/4 cup of this sauce for drizzling at the end; keep the rest for tossing.
Place the roasted vegetables and the roasted chickpeas in a large bowl. Pour the sauce (not the reserved 1/4 cup) over them and gently toss until evenly coated.
Divide 2 cups cooked brown rice between 6 serving bowls. On each bowl, layer 1/3 of the rice surface with 2 cups spinach divided among the bowls (so spinach covers about one third of the rice in each bowl), layer another 1/3 with the 1 cup bean sprouts divided among the bowls, and fill the remaining 1/3 of each bowl with the sauced roasted vegetables and chickpeas.
Drizzle the reserved 1/4 cup peanut sauce over the bowls, distribute any remaining peanuts as desired, and serve.