Combine potatoes and 1 to 1 1/2 quarts water in a 5- to 6-quart pan; bring to a boil over high heat, then cover, reduce heat, and simmer just until potatoes are tender when pierced, 15 to 20 minutes. Drain and return to pan. Shake gently over medium heat until liquid is evaporated, about 2 minutes. Remove from heat.
Add oil, vinegar, wine, shallots, 1 teaspoon salt, and pepper; mix gently.
Meanwhile, combine sausages and 1/3 cup water in a 10- to 12-inch frying pan over medium heat; cover and cook until sausages are firm to touch, about 10 minutes. Pour cooking liquid into a 1-cup glass measure; skim off fat and discard. You need 1/4 cup liquid; add chicken broth if necessary or discard remaining. Whisk mustard into measured liquid until smooth.
Turn sausages occasionally in pan over medium-high heat until browned, 5 to 7 minutes. Transfer to a board and slice diagonally about 1/4 inch thick.
Add sausages, warm cooking liquid, parsley, chives, and tarragon to potatoes; mix gently. Add salt to taste. Add arugula and mix gently. Pour into a large bowl. Serve warm.