Tear bread into pieces and soak in cold water for 5 minutes. Squeeze each piece very firmly until nearly dry. Residual water prevents proper emulsification.
Place the tarama in the bowl of a food processor or blender. Run the machine for 1-2 minutes before adding anything else, scraping down the sides once. The tarama will lighten and become smoother.
Add the squeezed bread and grated onion (if using) to the food processor. Process until the mixture comes together into a smooth, cohesive paste.
With the machine running, slowly drizzle in the olive oil alternating with lemon juice in thirds. If mixture separates, add 1 tsp cold water and blend.
Taste and adjust with more lemon juice for brightness, or cold water by the tablespoon if too thick. Tarama varies widely in saltiness.
Transfer to a bowl, cover, and refrigerate for at least one hour. Bring back to room temperature before serving.
Drizzle generously with extra virgin olive oil, scatter a few capers, and serve with torn bread.