To make the Dressing, Whisk 1/3 cup mayonnaise, 2 tablespoons water, 1 1/2 tablespoons dill, 1/4 teaspoon Kosher sea salt, and 3 tablespoons EVOO.
Boil the Eggs: Bring a medium pot of water to a boil. Gently lower the eggs and cook for 7 minutes for jammy centers. Transfer to an ice bath and chill for 5 minutes. Peel and quarter the eggs.
In a large bowl, combine mixed salad greens and fennel. Add the 3 tablespoons of reserved dressing and toss to coat.
Spread the dressed greens on a large serving platter. Layer on smoked salmon, quartered eggs, red onion slices, carrot ribbons, sliced garlic (if using), croutons, and mint sprigs.
Drizzle with remaining dressing. Sprinkle with the remaining 1/2 tablespoon of dill and 1/4 teaspoon sea salt. Top with grated Parmigiano Reggiano and freshly cracked black peppercorns (to taste).
For even more flavor, let the salad rest in the fridge for 10–15 minutes before serving.