Mincing shallots is not as essential a knife skill as, say, learning how to slice and dice an onion, but a properly minced shallot can spell the difference between a sweet, aromatic, and subtle vinaigrette and one that's overly pungent and onion-y.
See, like onions and garlic, shallots don't produce much of their strong, sulfurous aromas until their cells are ruptured and a number of precursor chemicals have a chance to react. Mincing a shallot cleanly and neatly can minimize that formation so that its sweeter, more subtle aromas can really shine through. Of course the first step to any fine knife work like this is to start with a very sharp knife, but once you've got that, mincing a shallot is not all that different from mincing an onion in miniature. It just takes a bit of practice.
I use the basic three-direction mince, starting by splitting the shallot in half lengthwise (if a single shallot has multiple bulbs, separate them first), laying it flat on its cut surface, and using a series of parallel vertical cuts, keeping it all connected at the root end. Next I make a couple horizontal slices, and final finish by making more vertical cuts at a 90-degree angle to my first cuts, dividing it neatly into a fine mince.
Because those internal precursor chemicals continue to react after the shallot is cut, make sure to mince your shallots just before you start cooking, especially if you're planning on using them raw. If you must mince in advance, keeping them nice and cold in the refrigerator will help slow down their funk.
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