How [can you] not specify the amount of food/drink(s) that you consume for stuff like burgers...?
#By saying “cheeseburgers.”
By saying “cheeseburgers.”
The things that don't require articles are uncountable, either naturally (water) or because you're talking about the substance (beef) as a substance and not as a specific instance. “I eat fried chicken” is talking about the substance ‘fried chicken’ and not about specific pieces of fried chicken.
There is no ‘cheeseburger substance’ (it's called ‘beef’ or ‘hamburger’) so you must use an article.
However, saying “I eat cheeseburgers” doesn't necessary imply that you have ever actually eaten any particular number of cheeseburgers in your life. It's just the way of speaking vaguely that it's something that's acceptable for dinner.
If you really needed to underline that you're speaking hypothetically, you'd change the verb: “I could eat a cheeseburger” (right now, for this meal) or “I could eat cheeseburgers” (generally, whenever) would both get that idea across.
Edit: Per your comments, “I could eat some fractional amount of a cheeseburger but not possibly a complete cheeseburger” is not an idea actual humans ever need to express (in any language) so there's no easier way to get that idea across that saying it all explicitly.