THE TOMATO COULIS: Grill the tomatoes over high heat (or roast on a baking sheet in a 500°F oven) until nicely browned. Peel and seed, reserving the peels. Fortify the brodo by simmering with the peels and some Parmigiano rinds, if available. In a blender, working in batches, combine the tomatoes, brodo, and vinegar. (Don’t fill the blender more than halfway.) Slowly add a small amount of the olive oil and butter to each batch, pureeing until incorporated. Combine the batches in a mixing bowl. Season to taste with salt and pepper. Set aside.
THE QUENELLES: In a large saucepan over high heat, combine the butter with 1¾ cups water. Bring to a simmer, then reduce the heat to medium. Add the flour and stir rapidly until fully incorporated. Transfer to a stand mixer fitted with a paddle attachment (or to a bowl, if using a hand mixer or whisk). Add the eggs and egg yolks, one at a time, mixing rapidly until each is incorporated. (If using a whisk, switch to a wooden spoon as the mixture stiffens.) The result is a classic pâte a choux dough, which can be refrigerated for up to one day until you are ready to use it.
Place the pike in a freezer for about 15 minutes to make it easier to cut. Dice the par-frozen fish and transfer to the bowl of a food processer. Puree until smooth. Pass through a fine-mesh sieve and combine with the heavy cream, kosher salt, and pâte a choux. Meanwhile, in a large saucepan, bring 1 gallon of water to a slow boil. Using 2 spoons of the same size, scoop and form a football-shaped quenelle. Repeat using the remaining fish mixture. Poach the quenelles, 2 or 3 at a time (so as not to overcrowd), in the boiling water until cooked through, about 3 minutes. Remove and drain.
Heat the oven to 500°F. Cut the bread into cubes and toss with olive oil. Toast on a baking sheet until browned. Transfer to a food processor and puree to make bread crumbs.
Place the cooked quenelles in an ovenproof baking dish. Cover each with tomato coulis and 1 thin slice of mozzarella. Bake until the cheese melts. Garnish each quenelle with 1 teaspoon of the bread crumbs and grated Parmigiano. Serve family style or on separate plates.