Preheat your oven to 400°F (200°C). Wrap the whole beets in aluminum foil and roast for 45-60 minutes until tender when pierced with a fork.
Once the beets are cool enough to handle, rub off the skins under cold water, cube them into bite-sized pieces, and set aside.
Season the chicken breast with half of the sea salt and black pepper.
Heat a grill pan or outdoor grill over medium-high heat and cook the chicken for 6-7 minutes per side until the internal temperature reaches 165°F.
While the chicken rests, shred the medium carrot using a box grater.
In a small bowl, whisk together the extra virgin olive oil, lemon juice, and the remaining salt and pepper to create the dressing.
In a large bowl, toss the arugula, shredded carrots, and roasted beets with the lemon vinaigrette.
Slice the grilled chicken into strips and place them over the salad bed.
Top the dish with crumbled feta cheese and serve immediately.