Preheat the oven to 425 degrees . On a rimmed baking sheet, toss the tomatoes, garlic and shallot with 3 tablespoons EVOO; season to taste with salt and pepper. Bake until the tomatoes burst, about 25 minutes. Mash the tomatoes with a potato masher directly on the baking sheet; set aside.
While the tomatoes cook, bring a large pot of water to a boil, salt it, add the gnocchi and cook until they rise to the surface and resemble fluffy pillows, 3 to 8 minutes; drain and set aside. Reserve the pot.
Season the fish with salt and pepper. In a large, ovenproof skillet, heat the remaining 2 tablespoons EVOO, 2 turns of the pan, over medium-high heat. Add the fish, skin side down, and cook, turning once, until firm to the touch, 2 minutes on each side. Transfer the skillet to the oven and cook the fish for 3 minutes longer.
In the reserved gnocchi pot, melt the butter over low heat. add the cooked gnocchi, tarragon and chives and toss well; season to taste with salt and pepper.
Top the fish with the mashed tomatoes and serve the gnocchi alongside.