Soak the seeds: Combine the brown mustard seeds, vinegar, and cold water in a glass jar or ceramic bowl. Avoid using aluminum or copper containers, as the acid in the vinegar will react with the metal and ruin the flavor. Cover loosely and let the mixture sit at room temperature for 24 to 48 hours. The seeds will absorb the liquid and double in size.
Add flavors & blend: Pour the plump, softened seeds and any remaining liquid into a food processor or blender. Add the kosher salt, turmeric powder, and sweetener if using. Pulse the mixture until you reach your preferred texture. For a traditional stone-ground texture, leave about a third of the seeds whole or coarsely cracked. For a smoother paste, blend for 2–3 minutes.
Adjust consistency: If the mustard feels too thick, blend in cold water one tablespoon at a time until it reaches a spreadable consistency. Keep in mind it will naturally thicken a bit more as it rests.
Age and mellow: Transfer your finished mustard into a clean glass jar with a non-metal or lined lid (vinegar can corrode bare metal). Place the jar in the refrigerator and let it rest for at least 3 to 5 days. The chemical compounds will stabilize, and the flavor will mellow out beautifully.