Whisk together 1/2 cup brine, mustard, and oil. Add more brine to taste, if desired, and season with salt and pepper. Dressing can be made up to 1 week ahead and stored in refrigerator; bring to room temperature before dressing salad.
Place potatoes in a large pot and cover with 2 inches of cold water. Add 2 tablespoons salt and bring to a boil over high heat. Reduce heat and simmer until potatoes are tender when easily pierced with a knife, about 18 minutes; drain. Return potatoes to pot and cook over high heat just until water has evaporated and potatoes are dry; remove from heat. When just cool enough to handle, cut potatoes in half and transfer to a large bowl along with dressing. Generously season with salt and pepper and gently fold to combine. Let mixture cool, tossing occasionally.
Add bread-and-butter pickles, cornichons, sugar snaps, and herbs. Gently toss to combine; season with salt and pepper.
Potato salad can be made up to 1 day ahead and stored in refrigerator; bring to room temperature before serving.