Break the broccoli into florets. Trim just the bottom off the stem, and then roughly chop all the stems. Pulse in a food processor until the broccoli is the size of rice. Depending on your food processor, you may need to do this in batches.

In a large skillet or sauté pan, heat 1 tablespoon olive oil over medium high heat. Add the mushrooms and sauté for 5 minutes until they start to brown. Stir in the balsamic vinegar, garlic, red pepper flakes, 1/2 teaspoon salt and 1/2 teaspoon pepper and continue cooking for a few minutes until the mushrooms are caramelized and fragrant. Transfer the mushrooms to a bowl and set aside.

Heat 1 tablespoon olive oil in the same sauté pan over medium high heat. Sauté the shallots, letting them soften, about 3-4 minutes, add the broccoli rice, 1 teaspoon salt and 1/2 teaspoon pepper. Cook the rice, stirring frequently, for 2-3 minutes, until it is warmed but still crisp. Fold in the mushrooms, grated Parmesan and parsley.

Top with shaved Parmesan before serving.