We have a cook who was injured while working at her OTHER job and
Has been out for 3 weeks now. She keeps going to the doctor and returning
With notes from the doctor saying she can't work. Legally, I don't think I
Can terminate her, but at what point do we start replacing her hours with an
Able body. Can we encourage her to use FMLA?

There are federal and state laws governing FMLA, so you'll need to check on your state's regulations. But because you know this is a medical issue (doesn't matter how the injury occurred, so long as it wasn't at your location, which would make it a workers comp issue), you can definitely give your employee the necessary paperwork to complete for FMLA, including doctor certification. FMLA means that you'll need to be able to return your employee to a comparable position when her doctor releases her, but in the meantime you can have someone filling in for her (another employee or a temp).