In this talk I generalize a result of Behrend-Fantechi and Baranovski-Ginzburg to the 1-shifted cotangent bundle of a smooth scheme over the complex numbers. More precisely, I show how twisted cotangent bundles arise as derived intersection of Lagrangians inside the 1-shifted cotangent bundle, and how quantizations of the twisted cotangent bundles arise as derived tensor products of quantized Lagrangians.