Is it safe to cut a bit into a "Double Header(?)" over a closet door that supports ceiling beams?
From International Residential Code, your local planning department is the final authority on what may or may not be done after inspecting what modifications you are intending to do. This is a job that requires a permit and consultation with an experienced contractor.

R802.7.1 Sawn lumber. Notches in solid lumber joists, rafters, blocking and beams shall not exceed one-sixth of the depth of the member, shall not be longer than one-third of the depth of the member and shall not be located in the middle one-third of the span. Notches at the ends of the member shall not exceed one-fourth the depth of the member. The tension side of members 4 inches (102 mm) or greater in nominal thickness shall not be notched except at the ends of the members. The diameter of the holes bored or cut into members shall not exceed one-third the depth of the member. Holes shall not be closer than 2 inches (51 mm) to the top or bottom of the member, or to any other hole located in the member. Where the member is also notched, the hole shall not be closer than 2 inches (51 mm) to the notch.

The tension side of members: In normal gravity loading, this is the bottom of the beam, in cantilever, the top of the beam.