Using a mandoline, slice Brussels Sprouts and carrots, using a cut-proof glove for safety.
Dissolve salt in a large bowl with 5 cups of warm water. Add cabbage, Brussels Sprouts, carrots, and green onion to the bowl and soak for 1 hour.
While the vegetables are soaking, prepare the chili paste. In a large bowl, stir together 1/4 cup water, rice vinegar, fish sauce, and sugar. Add chili flakes, ginger, garlic, and anchovies and blend thoroughly.
Drain vegetables using a sieve set over a bowl, reserving brined juices.
Add drained vegetables to the bowl with the chili paste and mix well to coat. Pack into glass jars (an old pickle jar works perfectly for this).
Pour some of the reserved brine into the bowl with the chili paste residue. Mix to capture leftover seasonings, and pour chili-brine into the jar. Cover and let stand at room temperature overnight. Refrigerate and eat within 1 week.