Remove and discard the outermost leaves of both cabbage heads. Cut each cabbage into quarters and remove and discard the cores. Finely shred the cabbage (a mandoline or sharp knife works well). Place the shredded cabbage in a large mixing bowl.
Peel and grate the 3 carrots and mince 4–5 large garlic cloves. Add the grated carrots and minced garlic to the bowl with the shredded cabbage.
In a medium saucepan combine 4 cups water, 1/2 cup granulated sugar, 2 1/2 tablespoons sea salt, 10 peppercorns, 5–6 whole cloves, and 4 bay leaves.
Bring the saucepan to a boil over medium-high heat, then reduce heat and simmer for 10 minutes.
After 10 minutes, stir in 1/2 cup olive oil and 2/3 cup white vinegar (5%). Remove the pan from the heat and let the mixture cool for 10 minutes.
Strain the liquid through a fine-mesh sieve into a bowl or measuring cup to remove the peppercorns, whole cloves, and bay leaves. Discard the solids.
Pour the strained liquid over the shredded cabbage, carrots, and garlic. Stir thoroughly to combine and distribute the liquid evenly.
Place a weight on top of the vegetable mixture to press it down while it cools (for example: a flat dinner plate that fits inside the bowl, then a heavy bowl of water on top of the plate). Leave the weight in place until the mixture is fully cool to the touch.
Pack the cooled sauerkraut tightly into a large jar, pouring any remaining liquid over it so the vegetables are fully submerged. Seal the jar and refrigerate for 24 hours before serving.