Prepare the Sauce:

1. Preheat the oven to 350 degrees (F). Quarter the tomatoes and lay out (cut side up–basically on one side) on the baking sheet.
2. Sprinkle on the minced garlic and torn basil. Drizzle over the olive oil and season generously with salt and pepper.
3. Using your hands, toss everything until well coated.
4. Roast for an hour and half, or until the tomatoes are completely soft and starting to collapse and caramelize.
5. Remove the tomatoes from the oven and allow to cool until almost room temperature.
6. To make the sauce, take several batches of the roasted tomatoes at a time and push through a fine-meshed sieve (using the back of a wooden spoon or rubber spatula). Discard the skin, seeds and fiber.
7. Before serving, reheat (and reduce as necessary–to thicken) and season with salt and pepper. At last minute, chiffonade another small handful of basil and add to the sauce for an extra boost of flavor.

For the Gnocchi:

1. Meanwhile, peel the potatoes, removing any blemishes as you go and cut into similar size chunks (1.5 inches roughly).
2. Add the potato chunks to a medium saucepan 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.
3. Using a potato ricer (this will yield a lighter, more fluffy gnocchi) or masher, mash the potatoes until smooth.
4. Transfer the mashed potatoes to another bowl and allow to cool down until lukewarm. Add the goat cheese, flour, salt, pepper, and egg (slightly beaten) and gently mix with your hands, until the dough begins to come together.
5. Season with salt and pepper and knead slightly to form a firm dough.
6. Cut the dough smaller pieces and roll each ball into 3/4 inch dowels. Cut dowels into 1 inch pieces and set aside on a lightly floured baking sheet or onto a piece of parchment.
7. To cook, bring a large pot of water to a gently simmer. Salt generously. Gently add the gnocchi, stirring gently, and as they rice to the surface, scoop them out with a slotted spoon (test one) and transfer to the pan of warm tomato sauce.
8. Serve immediately with parmigiano-reggiano cheese and top with basil (a fine chiffonade).