Arrange cherry tomatoes cut-side up on a large rimmed baking sheet. Drizzle olive oil over tomatoes and sprinkle with ½ teaspoon salt and a grinding of pepper. Roast tomatoes until they are somewhat puckered and shriveled, but still juicy, about 90 minutes.
Bring a large pot of water to a boil over high heat and salt generously.
In a large, deep sauté pan over medium-low heat, melt butter. When butter has just begun to foam, stir in shallot. Cook, stirring frequently, until shallot is softened but not browned, about 7 minutes.
Scrape in tomatoes and any juices that have collected on the baking sheet. Add thyme sprigs and pour in cream. Heat gently to a simmer over low to medium-low heat. Right before dressing the pasta, turn off heat and stir in basil.
Meanwhile, add pasta to boiling water and cook according to package instructions until al dente.
Drain pasta in a colander, reserving about 1 cup of pasta water. Return pasta to pot and stir in the cream tomato sauce. Add ½ cup of Parmigiano-Reggiano cheese. Toss gently to combine. Add 1/2 cup of reserved pasta water, if necessary, to loosen sauce and toss again.
I keep the remaining starchy pasta water in a container in the refrigerator. If you have leftovers, it works great to moisten the pasta to reheat the next day.