Correct Cabinet Hinges

I’m looking for some help in choosing the correct cabinet hinges. I built a few storage shelves in my basement and put trim around the openings hoping that I could put a door on them. However, I’m having a difficult time locating the correct cabinet hinges to use for the door. As you can see by the attached picture, the trim is completely covering the frame of the shelf. Every time I open the cabinet doors, the ousting of the door hits the trim and the doors won’t open even to 90 degrees. Can anyone help? If possible, I would like hinges that are surface mounted without having to drill a mortise opening. Thanks.