Combine garlic (or shallots) and 3 cups of the vinegar in a 2- to 2 1/2-quart pan over medium heat.
Simmer gently, uncovered, for 15 minutes.
Remove from heat and pour into a glass jar.
Cover and let stand at room temperature for 24 to 36 hours, then pour through a fine strainer into a bowl, pressing garlic or shallots with the back of a spoon to squeeze out as much liquid as possible; discard residue.
Measure liquid and add vinegar if needed to make 2 cups.
In a 5- to 6-quart pan, combine flavored vinegar, water, and sugar.
Bring to a rolling boil over medium-high heat.
Stir in pectin and bring to a boil that cannot be stirred down.
Boil, stirring constantly, for 1 minute.
Skim off foam, then spoon hot jelly into about 7 hot sterilized 1/2-pint canning jars, filling to within 1/4 inch of rims.
Wipe off rims; top with scalded lids, then firmly screw on bands.
Process jars in a hot water bath, let cool, and test seals as directed for Mango-Apricot-Date Chutney