This study presents an automated optimization-oriented strategy for designing high power amplifiers (HPAs) using deep neural networks (DNNs). The proposed strategy consists of two optimization phases that are applied sequentially. In the first phase, the circuit topology is optimized by determining the number of passive components in the input and output matching networks using deep learning classification network. In the second optimization phase, component values are estimated using a deep learning regression network with electromagnetic-based Thompson Sampling Efficient Multiobjective Optimization (TSEMO). The proposed approach is compact, in the sense that the optimum solution is automatically generated by the process, opposite to the conventional approaches where manual post-processing is required to prune the process outcomes. It addresses the problem of heavy reliance of the system performance on the designer's experience and automatically generates valid layouts. In the demanding HPA design problem, uses of DNNs have been shown to provide much more accuracy than conventional shallow neural networks. The effectiveness of the proposed method is verified by implementing two designed HPAs, including GaN HEMTs. The efficiency-oriented optimized amplifier reveals higher than 60% drain efficiency, and the gain-oriented optimized amplifier has 17.6-18 dB linear gain in the frequency band of 1.8-2.2 GHz.