Squeeze the water from the bread and place in the bowl of a food processor with the tarama. Process until combined. Add 2 tablespoons of the lemon juice and the optional smoked paprika and process to blend.
Drop in the yolk and pulse to blend. Very slowly, drizzle in the olive oil and process until the mixture is whipped, light in color and creamy. Add in additional lemon juice to taste.
At this point, if the mixture is too thick, add in some water, a little at a time until desired consistency is achieved. Remove mixture to a bowl and stir in the minced shallot.