Preheat oven to 350 degrees Fahrenheit. Toast pecans or walnuts for 10 minutes.
Raise oven temperature to 425 degrees Fahrenheit.
In a roasting pan or a cookie sheet, lay out brussels sprouts, trimmed and cut in half. Sprinkle with olive oil, sea salt, and pepper.
Roast brussels sprouts for 15 minutes. I like them crunchy on the outside, but still a bit crisp on the inside. If you like yours cooked well throughout, lower the cooking temperature to 400 and cook for 20-25 minutes.
In a sauce pan, add red rice, water, bouillon and butter. Over high heat, bring to a boil. Lower the temperature to low, and cover with a lid once water is boiling. Continue to cook for another 15-20 minutes, until all water is absorbed. Fluff rice with a fork and allow to cool a bit.
Mix all salad dressing ingredients together.
Once brussels sprouts and rice are cooked, combine the two in a bowl, and mix well with salad dressing.
On plates, layer arugula with rice and brussels sprouts mixture, then top with pomegranate arils and pecans or walnuts.