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Season the veal roast generously with kosher salt and freshly ground black pepper on all sides.
In a large Dutch oven or heavy pot, heat the extra-virgin olive oil over medium-high heat.
When the oil shimmers, sear the veal roast on all sides until well browned, about 2-3 minutes per side. Transfer the roast to a plate and set aside.
Add the chopped onion, carrot, and celery stalk to the pot. Sauté until softened, about 5 minutes.
Return the veal to the pot. Add the bay leaves, peppercorns, and 1 teaspoon of kosher salt. Pour in enough water to just cover the roast, then bring to a boil.
Reduce the heat to low, cover, and simmer until the meat is very tender, about 1 1/2 hours.
Remove the roast and let it cool completely in its cooking liquid. Once cool, wrap it tightly in plastic wrap and refrigerate for at least 4 hours or overnight.
For the sauce, combine the drained tuna, drained anchovies, capers, mayonnaise, and fresh lemon juice in a blender or food processor. Blend until smooth.
If the sauce is too thick, gradually add and blend in the reserved cooking liquid until the sauce reaches a smooth, pourable consistency.
Using a sharp knife or meat slicer, thinly slice the chilled veal roast against the grain. Arrange the slices slightly overlapping on a serving platter.
Spread the tuna sauce generously over the veal and garnish with additional capers.
Cover and chill for 30 minutes before serving to let the flavors meld.