What's the longest plausible duration for a total solar eclipse? is on hold as off-topic -- "This question does not appear to be about worldbuilding, within the scope defined in the help center". From the question:
Given a planet and moon roughly like Earth and our Moon, are there modifications we can make to the orbit, size, or rotational speed of either body that could extend this out, while still maintaining a stable system?
My target is 15 minutes for a single location on the planet to witness an eclipse, ideally while able to see the sun's corona. Preference is for the eclipse to be caused by a moon rather than a large planet.
How is this not about worldbuilding? Is it the title? (Since it doesn't really match the body I'll edit that now.)
A comment suggested asking on Astronomy, but the OP rightly pointed out that they'd close it as hypothetical.