The question that prompted this study was how teacher-researchers make meaning of ‘critical GenAI literacy’ through their professional education. To construct ‘critical GenAI literacy’ as the object of teacher-research, a problem-posing method was used to review the literature, establishing background knowledge of critical literacy and GenAI systems. The objective was to analyze teacher-researchers’ practices of ‘critical GenAI literacy’ while studying for a master’s degree in education. Focusing on the interconnectedness of teacher-researchers and GenAI systems, the method of relational socio-analysis was used to explore the multi-layered drivers for giving meaning to ‘critical GenAI literacy.’ The results indicate that teacher-researchers make meaning of ‘critical GenAI literacy’ by acquiring the know-how to use GenAI systems in their studies, sharing knowledge of the uses of GenAI systems in 汉语 (Hànyǔ, Hàn language), analyzing the contributions of GenAI systems to precarious employment, and evaluating the governing power of GenAI systems. A key outcome of this study is to show that teacher-researchers can work collaboratively to extend and deepen the meaning of ‘critical GenAI literacy’ by processing data and producing knowledge about the blurry and complex penetration of education and teachers’ work by GenAI systems.