I assume the shortcut is the one created at installation and will simply run FH with whatever settings you choose.
The File > Project Window list of Projects is chosen by the Location setting in that window.
The default Project that is opened needs one of those listed Projects to be selected and shown with a green tick.
i.e. Select your desired Project, probably ‘Michael & Barbara family’, then use More Tasks… > Set as Default Project.
That project’s GEDCOM file will also be specified in Tools > Preferences > Startup > Default Startup File.
You can delete any individual Project by using More Tasks… > Delete Project…
But that won’t delete the Family Historian Projects folder itself, which must be done using Windows File Explorer.