Tonnato Eggs

12 large eggs
1 (6-oz.) can oil-packed tuna
2 oil-packed anchovy fillets
1/2 cup mayonnaise
2 Tbsp. fresh lemon juice
2 Tbsp. capers, drained, divided
1/2 tsp. kosher salt
4 Tbsp. extra-virgin olive oil, divided
2 Tbsp. salmon roe (optional)

Bring a large pot of water to a boil over medium-high heat. Gently lower cold eggs into water. Cook 10 minutes, then drain and transfer to a large bowl of ice water.
Carefully peel eggs and cut in half lengthwise. (To get clean cuts, use a very sharp knife and wipe the blade between each cut.) Pop out yolks and transfer to a food processor. Arrange whites on a platter.
Add tuna, anchovies, mayonnaise, lemon juice, and 2 tsp. capers to food processor and purée until smooth. Add salt and 2 Tbsp. oil and pulse just until combined. Transfer to a pastry bag or resealable plastic bag and snip the corner. Pipe into egg whites.
Cook remaining 4 tsp. capers and 2 Tbsp. oil in a small skillet over medium heat, stirring occasionally, until capers are crispy, 3–4 minutes. Transfer to a paper-towel-lined plate and let cool.
Top eggs with crispy capers and salmon roe (if using).