stare intently
瞠 centers on fixing one's gaze with intense focus: staring intently, often with wide-open eyes.
瞠 combines 目 (eye) with 堂, which likely contributes the sound and may suggest a wide or open space, fitting the idea of staring with wide-open eyes.
The eye 目 on the left and the hall-like 堂 on the right suggest eyes opened as wide as a hall: staring intently.
For ドウ, imagine staring so intently that your eyes feel like they're in a daze: daze -> ドウ, and your wide eyes are fixed on the target.