Put the tuna and salt in a food processor and pulse until it’s blended. Add 1/3 cup mayonnaise and pulse until the ingredients are getting creamy. With the processor running, drizzle in the olive oil and lemon juice and process until the tonnato is very smooth and creamy. Taste and add more mayonnaise, olive oil, lemon juice, or salt. Store in the fridge for up to 1 week. Use extras as a spread on toast with hardboiled eggs and pickled red onions, or thinned with additional lemon juice and tossed with roasted potatoes.
For remaining dish: Preheat your broiler (or grill on high heat).
Spread all the broccoli on a rimmed baking sheet and broil — with no oil — until slightly softened and nicely charred on most surfaces, turning once. Check every five minutes or so. This could take as short as 5 minutes if you have a powerful broiler or up to 20 minutes if you do not. Alternatively, place the broccoli florets over high heat on the grill until well charred, flipping once.
Pile the still warm broccoli into a bowl and squeeze the half lemon all over it. Season with 1/2 teaspoon chile flakes and generous amounts of salt and black pepper. Add 1/4 cup olive oil and toss. Taste and adjust the seasoning until it’s delicious.
Arrange the broccoli on a serving platter, grate a nice shower of pecorino over the top, and serve with the lemon wedges. Set out the tonnato as a dip.