To prepare the salad dressing, place the chopped shallot in a small bowl and add the mustard, lemon juice and vinegar. Add a generous pinch of salt and several grinds of pepper. With a whisk, mix and then add the olive oil, continuing to whisk until emulsified. Taste and adjust the salt and pepper, oil or vinegar to balance the dressing to your liking.
Heat a frying pan over medium heat and toast the chopped walnuts a few minutes until they begin to brown. Remove from heat. Place arugula in a salad bowl and toss with the dressing. Add the walnuts, goat cheese and gently toss to distribute evenly.