How to fix a joist that has been cut out
Solution depends on what the other joists look like.  The joist before the one that has almost nothing left at the bottom could be ok.
The typical thing to do here would be to cut the joist that has nothing left, double block between the two side joists and sister those two sides joists so they are doubles and pickup the load from the cut joist.  The cut joist area then just becomes an opening in the floor structure.
You could certainly build a wall with a header below to support the joist past the cut area.  Height of header depends on span and if you are putting a door below it you are limited with door heights.
An structural engineer could probably prescribe something simpler.  I had a somewhat similar situation with 18' overspan joists and they were able to prescribe 3/4&quot; ply on each side 18&quot; past each edge of the cut.  My joist had a notch at the top but was mid span so the loading and stresses are different than your case.
I wonder if you could do something similar but girdle the joist and have the new joist scabs extend down an additional 3&quot; and block the inside of the girdle.  You'd loose 3&quot; of ceiling height for the girdle area but if you are putting up upper cabinets in this area you could probably just custom make the cabinets or have some kind of crown trim detail to hide the lowered portion or if this is a limited room just have the ceiling in this one room slightly lower.
Any solution outside of a support wall or doubling would require a structural engineer.