Sidecar

Sidecar is Apple's feature for using a compatible iPad as another Mac display. The iPad can extend the desktop or mirror it, subject to Apple's device, account and connection requirements. Apple’s Sidecar documentation

A Sidecar display is different from a virtual screen created by BetterDisplay. Set up Sidecar through Apple's workflow first; BetterDisplay's preferences do not replace that setup.

Once the display is detected, BetterDisplay's Connection Management includes preferences for how to treat it when disconnected. See General Settings for the related per-display settings.