In any recipe that requires sautéing, it's pretty likely that the instructions will include a step or some warning to ensure sufficient fat in the pan at all times. Our recipes frequently underscore the importance of adding more oil when the pan seems dry; Daniel makes this explicit in his Chicken Marsala and Osso Buco recipes, and Kenji notes it in the description of his eggplant caponata recipe and in his recipe for huevos rancheros.
It may seem like a throwaway bit of advice, but it's crucial for proper browning, and even those who you'd think would know better overlook its necessity. Just this weekend, in fact, as I was throwing a quick lunch together for a friend, I made the mistake of letting a pan of faux aloo tikki become too dry. I'd neglected to add more oil after flipping the spiced potato patties, and, while the first side was beautifully and evenly browned, the obverse ended up a little patchy, and about half the patties lost bits of their crusts to the bottom of my skillet.
Here's a firm reminder: Always* feel free to add more oil to a pan when sautéing if the pan seems dry, no matter how much oil the recipe calls for.
But see below for a caveat to this "always."
A consistent sheen of oil on the bottom of the pan promotes even cooking of your food and prevents sticking. The mechanics behind this are relatively simple: The points of direct contact between the food and the pan are heated through conduction (the direct transfer of heat between two solid bodies), while the parts of the food in contact with the hot oil are heated via convection (the transfer of heat between two solid bodies through the medium of liquid or gas). Because conduction is far more efficient than convection, those bits of food directly touching the pan tend to brown and cook faster. If your pan starts to run dry, too much of the food ends up cooking via conduction, which can result in burnt bits or spotty browning.
(Read this excerpt from Kenji's book for more information on the fundamentals of heat transfer.)
Oil can also help prevent sticking in two ways. At high enough heat, oil forms bonds with the few free metal atoms on the bottom of the pan, preventing them from bonding with proteins on the surfaces of food. (This is why establishing a layer of seasoning on cast iron and carbon steel pans is critical for making them nonstick.) Oil also acts as a buffer zone that allows proteins on the food's surface to cook adequately before they come in contact with the metal, thereby making them unavailable to bond with the surface of the pan.
Whenever you're sautéing anything, be sure to pay attention to the bottom of the pan. If it looks a little dry, don't be afraid to add more oil, and take particular care when sautéing in batches. And remember, this applies to anything, not just breaded cutlets and lean meats. Some vegetables soak up oil in ways that can't be accounted for; even a mirepoix can sometimes go inexplicably dry.
The only other caution we offer is to watch out for mushrooms and eggplant, since both tend to act like oil sponges and, when sufficiently cooked, can dump some of their absorbed oil back into the pan. There is no good rule of thumb for those cases, other than to observe them carefully. It's always best to maintain enough oil for even cooking and to prevent sticking, and if your eggplant or mushrooms expel a lot of oil, you may need to drain them slightly to prevent your final dish from becoming greasy.