Heat 1 tablespoon oil in a high-sided skillet over medium heat. Cook garlic until pale golden, about 3 minutes. Add 2 cups tomatoes and 1/2 teaspoon salt. Cook until soft, about 7 minutes. Add stock. Simmer until saucelike, about 1 minute. Add remaining cup tomatoes. Cook until warm, 1 to 2 minutes.
Meanwhile, bring a large pot of water to a boil. Cook pasta according to package instructions until al dente. Drain.
Toss pasta in skillet to coat. Stir in basil, reserving some for garnish.
Divide pasta among 4 plates. Top with any remaining sauce. Dot with ricotta, and drizzle with remaining teaspoon oil. Top with shaved cheese and remaining basil. Sprinkle with pepper.
This is really delicious, but the lasagna noodles are hard to deal with because they are so big. I recommend mafalda (mini-lasagna noodles) or farfalle. Breaking up the noodles before you cook them might also work well.