1Whisk brown sugar and mustard powder in a large bowl to combine. Add vinegar and honey; whisk well. Strain through a fine-mesh sieve into a large metal bowl. Add eggs and whisk until blended.
2Set bowl with mustard mixture over a large saucepan of gently simmering water (do not allow bottom of bowl to touch water; eggs may scramble). Cook, whisking and scraping bottom of bowl frequently, until mustard is thick and an instant-read thermometer registers 160°F, about 5 minutes.
3Divide mustard among jars. Screw on lids and chill.