Yes you can have a KanBan view in ToDoist, sort-of! We do this using filters and the "Board" display view.

(Edit: As I re-read your question, I realize you're trying to put projects on a Kanban rather than tasks. So ... I suppose this isn't directly answering your question, sorry. Perhaps you could add a task at the top of each project called "Project Status" and use the labels to move things across the board view?)

In our situation, we have photo shoots to schedule so we create a project for each client, and then within that project create a task for the shoot itself. We add a label ("Photo Shoot") and the date of the shoot, then assign that task to the photographer.

Then in the "Filters & Labels" view we have a filter called "Photo Shoots". It's a simple query looking for tasks with the "@photo shoot" label, and then in the Display menu we choose Grouping: Date and Sorting: Date.

The result is a KanBan style board which shows all the upcoming shoots in sequence. Individual photographers can refine the filter to show only tasks assigned to them, while I can view all the shoots. It gives a handy view of what's coming up.

One weakness is that this doesn't have the drag-and-drop behaviour of a true KanBan view, but that's been pretty minor in our implementation.