Preheat the oven to 425°F. Line a sheet pan with a piece of foil. Pat the chicken breasts dry and place onto the pan, skin-side up, with the red peppers alongside them, cut-side down. Coat both the chicken and peppers with olive oil on all sides. Season well with kosher salt and freshly cracked black pepper. Roast for 35 minutes on the top rack, or until the chicken is golden and cooked through.
Meanwhile, pour the walnuts into an oven-safe skillet in one, even layer. Roast for 5-6 minutes. Let cool. Roughly chop and set aside.
Make the vinaigrette by whisking the white wine vinegar, honey, and dijon mustard together in a small mixing bowl. Drizzle in both oils very slowly, whisking continuously until emulsified. Season with salt and pepper to taste.
Remove the chicken and peppers from the oven. Let cool until able to handle. Remove the meat from the chicken and shred into small pieces using your hands or two forks. Remove and discard the skins from the red peppers, and then cut the peppers into 1” pieces.
To assemble the salad, add the mixed greens, sliced fennel, sliced watermelon radish, olives, shredded chicken, red peppers, and chopped walnuts to a mixing bowl. Toss well while you drizzle in the vinaigrette to your desired amount.
Transfer the salad to a large salad bowl and garnish with the herbs, plus more freshly cracked black pepper.
I’ll typically make this salad just for myself and eat it over 3-4 days. To do so, prep the chicken, red peppers, and walnuts. Make the vinaigrette. Slice the fennel and watermelon radishes. Store all of the ingredients separately. Assemble each salad as needed with a little bit of every ingredient in a mixing bowl. Drizzle in enough vinaigrette to your desired consistency. Cut the herbs fresh and sprinkle over top.