Preheat your oven to 400°F and line a baking sheet with parchment paper.
Peel and cube the red beets into 1/2-inch pieces, then toss them on the baking sheet with 0.5 tbsp of olive oil, half the salt, and half the pepper.
Roast the beets for 25-30 minutes, or until they are tender and slightly caramelized on the edges.
While beets roast, season the chicken breast with the remaining salt, pepper, and fresh thyme.
Heat a grill pan or skillet over medium-high heat and cook the chicken for 6-7 minutes per side until the internal temperature reaches 165°F.
In a small bowl, whisk together the remaining 0.5 tbsp olive oil, balsamic vinegar, and Dijon mustard to create the dressing.
Let the chicken rest for 5 minutes before slicing it into thin strips.
Place the arugula in a large bowl, add the roasted beets and sliced chicken, drizzle with the balsamic dressing, and finish by sprinkling the crumbled feta on top.