Add basil, parsley, garlic, pine nuts, olive oil, lemon zest, and flaky salt to a food processor. Pulse until chunky, not smooth.
Bring a pot of salted water to a boil. Par-cook fingerling potatoes for 8 – 10 minutes until just tender. Drain and toss with olive oil and salt.
Pat the branzino dry. Season the cavity generously with salt and pepper.
Stuff each fish with half the pesto and half the potatoes. Pin closed with toothpicks or tie with twine.
Score the skin lightly on each side. Rub exterior with olive oil and salt.
Heat the grill to medium-high and oil the grates well. Branzino skin is delicate and can stick. It should sizzle on contact. If it goes on silent, let the grill heat longer.
Grill over medium-high heat for 6 – 8 minutes per side, until skin is charred and crisp and internal temp reads 130°F at the spine.
Combine shaved fennel, parsley, tarragon, dill, mint, and shallot in a bowl.
Dress with champagne vinegar, olive oil, flaky salt, and black pepper just before serving.
Lay salad as a base, place whole branzino over top, finish with a squeeze of lemon and drizzle of olive oil.