Pour the grappa and milk into an impeccably clean half-gallon glass jar with a tight-fitting lid. Scoop in the sugar, chocolate, and lemon. Cover the jar tightly and shake well to help the sugar begin to dissolve. It will look curdled; that's as it should be, and it's perfectly safe. Set aside in a cool, dark place and shake or stir with a clean spoon every day for 10 days.
Set a cheesecloth-lined colander over a bowl and pour in the mixture. When the mixture has finished draining, squeeze the cloth to release as much of the liquid as possible, and discard the solids.
Line a sieve with a paper coffee filter. Pour in the liqueur and let the mixture drip through the clean bowl—this can take up to 24 hours. Change the filter when it becomes clogged with the residue from the liqueur.
Pour the liqueur into a decanter with a tight-fitting top. It will keep at room temperature for up to six months.