In a large wok or saucepan over medium-high heat, warm 2 tablespoons of the oil, and sauté the onions and garlic with a pinch of the salt for about 5 minutes until soft and translucent.
Transfer half of the onion and garlic mixture to the blender, and add 1 teaspoon of the salt, vegetable broth, canned tomatoes (and their juices), sun-dried tomatoes, basil, thyme, oregano, sweetener, and red pepper flakes and blast on high for 30 to 60 seconds until the sun-dried tomatoes are pulverized.
Throw the cauliflower into a food processor fitted with the S blade, and pulse until you have very tiny pieces the consistency of rice or ground beef. (You may have to do this in two batches; see the Notes.)
Turn on the pot with the remaining half of the onion and garlic mixture to medium heat. Add the cauliflower “rice” to the pot, and sauté for about 10 minutes until just cooked through. Add the mushrooms, and the remaining tablespoon of oil, and sauté for about 1 to 2 minutes, until just softened.
In a small bowl, mix the tomato paste, balsamic vinegar, and mustard together until well combined and add this mixture to the cauliflower-mushroom mixture until well combined and evenly coated.
Add the blended tomato mixture and the remaining teaspoon of salt and simmer, uncovered, stirring every few minutes for 15 to 20 minutes, until the sauce has thickened and slightly reduced, and the flavors have meshed. Season with salt and pepper, to taste. You should have about 6 cups (1.4kg) of sauce.
Serve immediately, or divide into 1 cup (250g) portions, and store according to the instructions in the recipe notes.