To pickle the vegetables: In a large non-reactive saucepan, bring the water, vinegar, sugar, and salt to a boil, until the sugar and salt are dissolved.
Meanwhile, place the carrots and cauliflower in a large bowl (or two separate bowls if you don't want the color of the carrots to tint the cauliflower). Pour the pickling liquid over top. Press with a spatula to submerge. Transfer the vegetables and liquid to storage containers — use separate containers if you don't wish the cauliflower to turn color. Let cool to room temperature, then refrigerate. Store for weeks if not longer.
Make the dressing: Place the shallot and garlic in a large bowl. Cover with the vinegar, the juice of half a lemon, and a pinch of salt. Let stand 10 minutes. Add the anchovy and the flesh of the avocado. Mash together with a fork or whisk. Add the herbs. Gradually whisk in the olive oil. Taste. You can stop here if you wish or add the cream for a richer taste. Taste and adjust the seasoning to taste with salt (1/4 teaspoon at a time), pepper, lemon or vinegar if necessary.
To serve: Arrange pickled vegetables and other vegetables on a platter. Spoon green goddess dressing into a dip bowl and serve alongside the vegetables.