Adjust the thickness of your mandoline slicer. The target thickness should be around the thickness of a Ritz cracker. Cut one slice at a time until you dial in the thickness. Once you've got it, work quickly but carefully to end up with a bowl of uniformly thick shallot rounds. Using the hand guard is a valid option here.
Pour the cooking oil into a deep cooking vessel. Heat over medium-high heat until the oil shimmers. If using a thermometer, it should register 325ºF.
Set your strainer over a mixing bowl, and line a half hotel pan with paper towels.
Working in batches, carefully add a couple of handfuls of shallots at a time. The oil will bubble aggressively. Cook the shallots until they turn pale golden, stirring occasionally. Transfer the shallots and cooking oil through the strainer, shake to drain, then move the shallots to the prepared hotel pan. Season with a pinch of kosher salt. Pat dry with additional paper towels. Return the oil to the pot between batches and repeat.
The shallots will continue to cook and crisp. They can be stored at room temperature in a sealed container for a few weeks. Adding a desiccant will help maintain crispiness.