We describe a framework to construct a perturbative quantization of nonlinear AKSZ Sigma Models on manifolds with and without boundary, and show that it captures the change of the quantum state as one changes the constant map around which one perturbs. Moreover, we show that the globalized quantum state can be interpreted as a flat section with respect to a flat connection. This flatness condition is a generalization of the modified Quantum Master Equation as in the BV-BFV formalism, which we call the modified "differential" Quantum Master Equation.
