What's the best way to connect a high-resolution external display to a MacBook Air M3?

Use USB‑C directly. It supports up to 6K at 60Hz via DisplayPort alt mode. HDMI adapters cap out at 4K 60Hz because they’re HDMI 2.0, and the M3 Air doesn’t support HDMI 2.1 over USB‑C. So skip the MacBook Air adapter for crisp, big monitors.