Many user interfaces involve attention shifts between primary and secondary tasks, e.g., when changing a mode in a menu, which detracts the user from their main task. In this work, we investigate how eye gaze input affords exploiting the attention shifts to enhance the interaction with handheld menus. We assess three techniques for menu selection: dwell time, gaze button, and cursor. Each represents a different multimodal balance between gaze and manual input. We present a user study that compares the techniques against two manual baselines (dunk brush, pointer) in a compound colour selection and line drawing task. We show that user performance with the gaze techniques is comparable to pointer-based menu selection, with less physical effort. Furthermore, we provide an analysis of the trade-off as each technique strives for a unique balance between temporal, manual, and visual interaction properties. Our research points to new opportunities for integrating multimodal gaze in menus and bimanual interfaces in 3D environments. CCS CONCEPTS • Human-centered computing → Human computer interaction (HCI); Virtual reality; Mixed / augmented reality.