How to Make It
In a medium saucepan of boiling salted water, cook the carrots until just tender, about 4 minutes. Using a slotted spoon, transfer the carrots to a colander and refresh under cold water. Add the potato slices to the boiling water and cook until just tender, about 4 minutes. Using a slotted spoon, transfer the potatoes to a large plate. Reserve the cooking water.
In a large bowl, combine 1/4 cup of the olive oil with the lemon juice, capers, dill and chives. Add the carrots and potatoes and season with salt and pepper. Let stand at room temperature for 1 hour.
Bring the cooking water back to a boil. Add the sausages and simmer over moderate heat until just cooked through, about 5 minutes. Transfer the sausages to a plate.
In a small bowl, combine the remaining 3 tablespoons of olive oil with the mustard, sherry vinegar, shallots and thyme and season with salt and pepper.
Melt the butter in a medium cast-iron skillet. Add the sausages and cook over moderately low heat until browned all over, about 8 minutes.
Slice the sausages. Mound the sausages and potato salad on plates. Drizzle 2 teaspoons of the mustard dressing over each plate and serve.