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.