Preheat the oven to 400°F. Peel the carrots and cut into batons. Transfer to a baking sheet and drizzle with olive oil. Season with kosher salt and pepper and toss to coat. Roast for 25 to 30 minutes until tender. Let cool for a few minutes until the carrots are warm, but not hot, before using in order to prevent the dressing from melting.
In a small blender or food processor, combine 1/4 cup yogurt, 1/4 cup feta cheese, mayonnaise, the chopped fresh dill and the minced garlic. Add the juice from one half of the lemon. Reserve the other half lemon for the arugula. Add the kosher salt, and pepper. Blend until the dressing is smooth.
Transfer the dressing to a bowl and whisk in the remaining yogurt and feta cheese. Whisking in the remaining yogurt will help to thicken the dressing. Chill until ready to serve.
While the carrots are roasting, prep the arugula salad. Toss together the baby arugula and fresh dill sprigs. Squeeze the remaining half lemon over the greens and drizzle with a little olive oil. Sprinkle the salad with kosher salt and toss together.
To serve, arrange the warm roasted carrots on a large serving platter and drizzle with the feta dill yogurt dressing. Top with the arugula dill salad and enjoy!