Preheat your oven to 400°F and line a small baking sheet with parchment paper.
Dice the red potato into half-inch cubes and toss with half of the olive oil and a pinch of sea salt.
Spread the potatoes on the baking sheet and roast for 20-25 minutes until they are golden and tender.
Season the chicken breast with salt and pepper, then grill over medium-high heat for 6-7 minutes per side until the internal temperature reaches 165°F.
Bring a small pot of water to a boil, carefully lower the egg in, and cook for 7 minutes for a jammy yolk; immediately transfer to an ice bath, then peel and halve.
In a small jar or bowl, whisk together the remaining olive oil, lemon juice, and Dijon mustard to create a smooth dressing.
Place the fresh arugula in a large bowl and top with the sliced grilled chicken, warm roasted potatoes, and the halved egg.
Drizzle the lemon vinaigrette over the salad and enjoy immediately.