Prepare Sauce: Preheat the oven to 350°F (175°C) with a rack in the center position. Lay the tomato pieces, cut side facing up, on a half sheet pan, distributing them evenly across the surface.
Sprinkle on the minced garlic and ¼ cup of the torn basil. Drizzle with the olive oil and season generously with salt and pepper. Using your hands, toss everything until well coated.
Roast for 75 to 90 minutes or until the tomatoes are completely soft and starting to collapse and caramelize. Remove the tomatoes from the oven and cool until lukewarm. To make the sauce, take several spoonfuls of the roasted tomatoes and push through a fine-meshed sieve (using the back of a wooden spoon or rubber spatula). Discard the skin, seeds and fiber. Note: Alternatively use a food mill.
Before serving, reheat in a wide skillet on the stove. Reduce to thicken if desired and season with salt and pepper to taste. Just before tossing with the gnocchi, stir in the remaining torn basil.
Prepare Gnocchi: As the tomatoes are roasting, peel the potatoes, removing any blemishes as you go and cut into similar size (roughly 1½-inch) chunks.
Add the potato chunks to a medium pot and fill with cold water. Salt the water generously and bring to a boil. Lower the heat and simmer until the potatoes are completely tender, then drain well. Using a potato ricer (this will yield a lighter, more fluffy gnocchi) or masher, press and mash the potatoes into a large mixing bowl. Allow the mashed potatoes to cool down until lukewarm.
Once the potatoes have cooled slightly, add the goat cheese, flour, salt, pepper, and egg (slightly beaten) to the bowl. Mix everything together with your hands until the dough begins to come together. Season with salt and pepper and knead slightly to form a firm dough.
Cut the dough into portions and roll each dough portion into ¾-inch wide dowels. Cut the dowels to create 1-inch wide gnocchi pieces. Set the gnocchi on a lightly floured baking sheet or onto a piece of parchment paper.
Cook the Gnocchi: Bring a large pot of water to a gently simmer. Salt generously. Gently add the gnocchi, stirring gently. Once they rise to the surface and are cooked through, scoop them out with a slotted spoon and transfer to the pan with the warm tomato sauce. Serve immediately with grated parmigiano-reggiano cheese for serving.
Adapted, just slightly, from Hugh Fearnley-Whittingstall’s River Cottage Veg.