The best would be to have doors that give you information about what's inside the fridge. Here are some examples of such materials and technologies:
Switchable Smart Glass Vision Panels
Easily integrable into any door and could be installed on the front face of a thick plexiglass fridge door for the desired purpose. The touch of the handle would activate the transition between a non-visible (mirror or iced) and a visible (completely transparent) look. The downside is that it could be tricky to design it so that you can have something in the fridge door like milk or a bottle of juice.
Dark glass door with backlight
Simple tinted glass door could be installed as a fridge door. Because of the tint, the food wouldn't be easily noticable, especially from the distance. In that way, your fridge would keep that designer look. However, when you grab a handle, backlight turns on enabling you to see what there on the fridge shelves.
This product already exists and could ne used for the desired purpose. However, to match the idea description from Michaela D's contribution, the camera should have a screen mounted on the front face of the fridge door. Simple touch of the handle or a screen would turn it on enabling you to see what's left to eat. This approach would also allow you to gather information for the checklist, or from the product labels and use it to notify you about missing or expiring food and drinks.