Cook the pasta in generously salted water until al dente. Reserve ⅓ cup of the cooking water, then drain.
In a wide skillet over medium heat, melt the butter with the olive oil. Add the shallot and cook, stirring occasionally, until soft and translucent, about 3 minutes. Add the garlic and cook until fragrant, about 30 seconds.
Add the tomatoes and season lightly with salt. Cook, stirring occasionally, until the tomatoes collapse and the mixture becomes jammy, 8 to 10 minutes. Stir in the cold butter until fully emulsified.
Transfer the mixture to a blender or food processor and blend until completely smooth. Return the sauce to the skillet over low heat. Add the reserved pasta water and stir to loosen. Taste and adjust seasoning.
Add the pasta, basil, and Parmigiano Reggiano, tossing until evenly coated. Serve immediately with burrata torn over the top and chili flakes, if using.