Heat the olive oil in a large skillet or Dutch oven over medium heat. Sauté the onion until it starts to soften, about 5 minutes, then add in the mushrooms and asparagus and cook for 5 more minutes.
If you haven't purchased prepared cauliflower rice for convenience, this would be a good time to to prepare the rice while the vegetables are sauteeing. Simply break up a head of cauliflower into florets and then pulse them in a food processor, or grate them with a box grater, to create a rice-like texture.
Add in the garlic, balsamic vinegar, and thyme and sauté for another minute. Add in the prepared cauliflower rice, coconut milk, and 1 teaspoon of salt, then stir well.
Bring the coconut milk to a simmer and cover the pot with a lid. Lower the heat and let the vegetables cook until the cauliflower rice is tender, about 10 minutes. Remove the lid. If any excess liquid remains at the bottom of the pan, you can raise the heat to simmer it off. Adjust any seasoning to taste, adding more salt if desired. (I added another 1/2 teaspoon to mine-- I find that cauliflower needs quite a bit of seasoning!)
Serve warm with sliced green onions and black pepper on top. Leftovers can be stored in an airtight container in the fridge for up to 4 days.