Heat 1 tablespoon of olive oil in a large pot over medium-high heat. Add onion, garlic, cabbage and sprinkle with sea salt. Sautee until the veggies have softened, about 5-7 minutes.
Scoop about half of the mixture out into the blender. Let cool slightly. Add milk, half and half and fresh dill to the mixture in the blender. Blend until smooth.
Add the blended mixture back into the rest of the soup in the pot. Stir and allow to come to a simmer and heat through completely. Serve hot and garnish with extra dill if desired.