Heat the olive oil in a large soup pot or Dutch oven over medium to medium-high heat. Sauté the onions for about 5 minutes, or until they begin to soften and brown a bit on the edges.
Add the chicken, and cook for about 5 minutes, browning on both sides.
Add the tomatoes with their juice, the water, thyme, sugar, salt and pepper. (I adjusted over the course of cooking and found a total of 1 1/2 teaspoons kosher salt and a scant half teaspoon of fresh pepper to be perfect for our tastes.)
Bring the mixture to a boil, then immediately reduce the heat to low, and simmer, uncovered, for 45 minutes.
At this point, I removed the chicken to a plate and half shredded, half cut it into bite-size pieces, and then returned to the pot.
Finally add the chicken broth, bring to a boil, then add the brown rice. Cover and continue cooking over low heat (you want the mixture to be at a very gentle simmer) for another 45-50 minutes or until the rice is cooked but is not mushy. (Check early a time or two as the rice will continue to cook in the hot broth for a few more minutes after being removed from the heat.)
Stir in the chopped spinach, the balsamic vinegar, and check for seasoning.
Serve with a sprinkling of cheese, if desired.