Heat the oven to 425ºF.

• Cut the fennel lengthwise into quarters and cut away the core; cut the quarters lengthwise into ¼-inch-thick slices.
• Scrub or peel the carrots and trim off the tops; cut the carrots on the diagonal into ½-inch-thick slices.
• Peel and cut the red onion crosswise into ¼-inch-thick slices; separate the slices into rings.

On a sheet pan, toss the fennel, carrots, and onion with 1 to 2 tablespoon oil and season with salt and pepper. Spread in an even layer and roast, stirring occasionally, until the vegetables are lightly browned and tender, 18 to 20 minutes.
While the vegetables roast, prepare the chicken.

• Pat the chicken dry with a paper towel; season generously with salt and pepper.
• Cut half the lemon into 4 slices; juice half and set aside for the dill sauce.

Using your fingertips, loosen the skin of each chicken breast. Tuck 2 lemon slices under the skin and gently stretch the skin to cover the lemon slices.
In a medium frying pan over medium-high heat, warm 1 to 2 tablespoons oil until hot but not smoking. Add the chicken, skin side down, and cook until the skin is well browned, 5 to 6 minutes. Turn and cook until the meat is firm and cooked through, 3 to 5 minutes, depending on thickness.
While the chicken cooks, prepare the dill sauce.

• Coarsely chop the dill.

In a small bowl, stir together the dill, lemon juice, and 1 to 2 teaspoons oil. Season to taste with salt and pepper.

• Cut the olives in half.
• Strip the mint leaves from the stems; coarsely chop the leaves.

Add the olives and mint to the roasted vegetables and stir to combine.

Transfer the chicken and vegetables to individual plates. Drizzle the dill sauce over the chicken and serve.