Our back door has 12 diamond patterned glass panes on the top 1/2 of door. A modified dutch door style in one piece. One of them is broken. My dh says we need to take the door to the glass place for replacing as there is no putty. This piece of glass is maybe 7 in. @ the widest/longest. Not double glass. what you say?
Are these panes separate - true "divided lights"? Are the panes flat or beveled?
If they are separate panes and installed with wood dividers, there should be a way to disassemble the whole window, then separate the mullions around the broken piece and replace just that one.
This is not impossible for a handy-person, but might be a whole lot quicker (and maybe cheaper) if done by a professional. Take a picture then contact some local glass shops and ask when showing them the picture. Ask price and turnaround time.
When you find one you loke, take the door to them. Hopefully they can fix while you wait.