Nanopore sequencing generates noisy electrical signals that need to be converted into a standard string of DNA nucleotide bases using a computational step called basecalling. The accuracy and speed of basecalling have critical implications for all later steps in genome analysis. Many researchers adopt complex deep learning-based models to perform basecalling without considering the compute demands of such models, which leads to slow, inefficient, and memory-hungry basecallers. Therefore, there is a need to reduce the computation and memory cost of basecalling while maintaining accuracy. Our goal is to develop a comprehensive framework for creating deep learning-based basecallers that provide high efficiency and performance. We introduce RUBICON, a framework to develop hardware-optimized basecallers. RUBICON consists of two novel machine-learning techniques that are specifically designed for basecalling. First, we introduce the first quantization-aware basecalling neural architecture search (QABAS) framework to specialize the basecalling neural network architecture for a given hardware acceleration platform while jointly exploring and finding the best bit-width precision for each neural network layer. Second, we develop SkipClip, the first technique to remove the skip connections present in modern basecallers to greatly reduce resource and storage requirements without any loss in basecalling accuracy. We demonstrate the benefits of RUBICON by developing RUBICALL, the first hardware-optimized basecaller that performs fast and accurate basecalling. Compared to the fastest state-of-the-art basecaller, RUBICALL provides a 3.96x speedup with 2.97% higher accuracy. We show that RUBICON helps researchers develop hardware-optimized basecallers that are superior to expert-designed models.