Recognizing the cognitive workload (CW) of operators is crucial to avoid human factor failures. Recently, multimodal CW recognition has attracted increasing attention since it leverages complementary information from different physiological modalities to enhance CW recognition performance. However, in real-world scenarios, not all modalities are always available. The performance of multimodal CW recognition may degrade when any modality misses, especially electroencephalogram (EEG). Although existing methods are capable of addressing the issue by explicitly recovering the missing modalities based on the available ones, they struggle to generate high-quality missing modality features due to their neglect of inter-modality relationships. In this paper, we propose a novel multimodal learning framework for CW recognition to address the incomplete modality problem. To recover the missing modality, a mutual information-assisted recovery strategy, which can maximize the mutual information between the missing and the available modalities, is used to train a feature generation module. Furthermore, to efficiently utilize complementary multimodal information, we employ a feature fusion strategy based on a channel attention mechanism to help the model focus on the key information. As a result, the proposed framework can achieve good CW recognition performance in missing modality scenarios. Our method attains an average accuracy of 75.04% on a public dataset, which is the highest among all compared methods and demonstrates the effectiveness of our framework.