Wash and chop all the vegetables, and put them into a food processor with the tomato puree, tarragon, anchovies, salt, pepper, and lemon juice. Add two tablespoons of water, and process the mixture into as fine a paste as you can.
Add the vinegar to the mixture, give it a short pulse in the processor, and pour the resulting slurry into a bowl.
Slowly whisk the olive oil into the mixture, pouring it in a thin stream until completely blended.
