In this paper, we introduce a new class of parameterized controllers, drawing inspiration from Model Predictive Control (MPC). These controllers adopt a Quadratic Programming (QP) structure similar to linear MPC, with problem parameters being learned rather than derived from models. This approach may address the limitations of commonly learned controllers with Multi-Layer Perceptron (MLP) architecture in deep reinforcement learning, in terms of explainability and performance guarantees. The learned controllers not only possess verifiable properties like persistent feasibility and asymptotic stability akin to MPC, but they also empirically match MPC and MLP controllers in control performance. Moreover, they are more computationally efficient in implementation compared to MPC and require significantly fewer learnable policy parameters than MLP controllers. Practical application is demonstrated through a vehicle drift maneuvering task, showcasing the potential of these controllers in real-world scenarios.