To make the pesto: Grind the basil, pine nuts and grated Parmesan cheese with a mortar and pestle until a smooth paste forms. Slowly add 1/4 cup/50 ml of olive oil, grinding until a smooth sauce forms. If you don't have a mortar and pestle, use a food processor instead. Season the pesto to taste with salt and pepper. Cover the pesto and set it aside.
To prepare the pasta and shrimp: Preheat the oven to 450°F/230°C Place the vine of tomatoes in an ovenproof skillet. Drizzle 1 tablespoon/15 ml of oil over the tomatoes then sprinkle the tomatoes with salt and pepper.
Roast the tomatoes in the oven until they are heated through, about 8 minutes. Meanwhile, bring a large pot of salted water to a boil over high heat. Add the linguine to the boiling salted water and cook until al dente, stirring occasionally to prevent it from sticking, about 2 minutes.
While the linguine cooks, heat the remaining 3 tablespoons/45 ml of olive oil in a medium sauté pan over medium heat. Add the garlic and shrimp and sauté for about 3 minutes or until the shrimp are just cooked through and the garlic is tender.
Add the pesto to the shrimp mixture. Drain the linguine, reserving about 1 ½ cups/355 ml of the cooking liquid. Toss the linguine in a large bowl with the shrimp and pesto mixture like you've never tossed before in order to coat the linguine and shrimp with the pesto, adding enough of the reserved cooking liquid to moisten the sauce so that it coats the pasta evenly.
Using a two-pronged carving fork, swirl some of the pasta around the fork. Slide the pasta off the fork and mound it in the center of a plate. Repeat for each serving. Arrange the shrimp and roasted tomatoes around the pasta. Garnish with shaved Parmesan cheese and serve. That'll do it!