The electromagnetic-inverse-scattering (EMIS) problem is solved by a novel two-step deep-learning (DL) approach in this article. The newly proposed two-step DL approach not only predicts the multifrequency EM scattered field, but also overcomes the limitation of the conventional methods for solving EMIS problems, such as expensive computational cost, strong ill-conditions, and invalidity on high contrast. In the first step, the complex-valued deep residual convolutional neural network (DRCNN) is utilized to predict multifrequency EM scattered fields only using single-frequency EM scattered field information. Based on a new complex-valued deep convolutional encoder-decoder (DCED) structure, the second step utilizes the obtained multifrequency EM scattered field "images" to realize the reconstruction of the target scatterers. In such a manner, the proposed approach can solve the EMIS problem accurately and efficiently even for inhomogeneous and high-contrast scatterers. The training of the proposed two DL models is based on the simple synthetic dataset. Numerical examples based on various dielectric objects are given to demonstrate the accuracy and performance of the newly proposed approach. The proposed DL-based method opens a new path for handling real-time quantitative microwave imaging.