If I understand your question correctly, you've got users, and users have a list of things. A lot of things. And they're all the same kinds of things.
If I, as a user, click on a button that says "show my things" I want it to show my things. I want it to show ALL the things!
Sersiously though, you can't NOT show anything if the button promises to show things. If it's possible to show everything at once, ie. performance wise, that would be preferable. I don't know what kind of things these are, but perhaps you can organize them somehow. If they're identifiable by name, you could opt for an index-like layout that makes it easy to scan alphabetically. You could go for a list with the letters of the alphabet on top so I can for instance click the P and quickly skip to that section. It might make sense to have 5 columns of things and that might make it easier. Perhaps I can just hit CRTL-F and find what I'm looking for with that. And you can always add search functionality on top of that if you need to.
If you can't show everything, like if you need to use a select-box, or performance will not allow it, show at least something. Show the 5 most accessed things, the 10 most recently accessed. Whatever makes sense for your application. The first X things by alphabetical order hardly ever makes practical sense, make it a meaningful selection.
Then, if I don't find what I'm looking for, I can start entering a search string to get a different selection of things to pick from.
Infinite scrolling doesn't allow me to use CTRL-F. It also doesn't allow me skip down by giving the scrollwheel/trackpad/touchscreen a wipe as I'll probably bump into the limit of what you've already loaded. It sounds like your users will at least kind of know what they're looking for and not casually browsing the list (in which case infinite scrolling may be fine). Give them the option to tell your system something about what they're looking for and narrow down the set of things for them.