In the base of a blender or food processor fitted with the blade attachment add the olive oil, mayonnaise, lemon juice, tuna, anchovies, capers, lemon juice and 1 tablespoon (15g) cold water.
Blend until the sauce is smooth. Season to taste with salt. The tonnato sauce can be made up to 2 days in advance and stored refrigerated in an airtight container.
For the broccolini:
In a large pot over medium heat, bring 3 quarts (2.8L) of heavily salted water to a rolling boil. Have a large bowl of ice water ready on the side. Add the broccolini and cook until bright green and tender, about 2 minutes.
Drain the water and place the broccolini in the ice bath to stop the cooking process. When the broccolini is cool remove from the ice bath and pat dry.
Preheat the broiler to high with a rack set in the top third of the oven. Arrange broccolini on an aluminum foil-lined baking sheet. Drizzle the broccolini in the olive oil, tossing to coat then arrange in an even single layer.
Season with salt and pepper then broil until the tips of the broccolini start to char but are not burnt, about 5 to 6 minutes.
Arrange broccolini on a baking sheet and drizzle with ¼ cup of the tonnato sauce. Garnish with parsley leaves.