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Heat the oven to 300 degrees F. Spread the cubed bread out on a rimmed baking sheet. Toasted until bread is crisp and fragrant, 20 - 30 minutes, depending on the staleness of the bread you're using.
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Place toasted bread, plus any crumbs from the baking sheet in a large mixing bowl. Grate a third of your tomato stash over the bread. The tomato flesh will easily give way, leaving just the peel in your hand. Discard the peel. Toss bread and tomato mixture with a generous pinch of sea salt and set aside for 15 minutes.
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To make the vinaigrette, mash the anchovies and garlic into a paste. Add a pinch of sea salt, the vinegar, and the Dijon. After 5 minutes, slowly whisk in the olive oil. Taste and add sea salt as needed.
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Chop the remaining tomatoes and toss with the soaking bread (and maybe another pinch of sea salt). Place the cukes and onions in their own little bowls and toss with a sprinkle of sea salt and a teaspoon of vinaigrette.
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When ready to assemble, gently fold the vegetables in with the tomato-soaked bread. Toss everything with a generous drizzle of vinaigrette. Add the herbs, black pepper, and mozzarella, if using. Taste and add more sea salt and a big drizzle of olive oil to taste. Serve right away.
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Leftover salad will keep surprisingly well in the fridge overnight, but the flavors are best day-of.