Step 1 Cut an X in the stem end of each Brussels sprout and remove any outer leaves that are discolored or loose. In a covered pot, steam the sprouts over rapidly boiling water until they are tender, about 8 to 10 minutes. Cut each sprout into lengthwise quarters and set aside. (The Brussels sprouts can be prepared up to this point 8 hours in advance and refrigerated, tightly covered.)
Step 2 Render the bacon in a dry skillet over medium heat until it softens and releases its fat, about 5 minutes. Add the minced shallot and cook just until fragrant, about 1 minute. Add the chestnuts and chicken broth to the pan, cover and simmer until the chestnuts are quite tender and sweet, about 10 minutes.
Step 3Add the vinegar and Brussels sprouts and cook, letting the flavors marry, about 5 minutes. Season to taste with salt and pepper and add a little more vinegar, if necessary, to brighten the flavors.