Preheat the oven to 350 degrees. Arrange 3 racks evenly spaced in the oven.
Place the chicken, skin side up, on a sheet pan. Rub with olive oil, sprinkle with salt and pepper, and roast on the top rack for 35 to 40 minutes, until the internal temperature is 140 degrees. Set aside to cool, discard the skin and bones, and shred the chicken by hand into large, roughly 1 x 4-inch, pieces.
Place the bacon on a baking rack on a second sheet pan and roast below the chicken on the second oven rack for 20 to 30 minutes, until browned. Remove to a plate lined with paper towels to cool. Cut in large dice. On a third sheet pan, combine the almonds, cashews, and walnuts and roast under the bacon for 10 to 12 minutes, tossing once, until lightly browned. Set aside to cool.
In a medium bowl or 2-cup measuring cup, whisk together 2/3 cup of olive oil, the vinegar, mustard, honey, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper, and set aside.
In a large serving bowl, combine the frisee, kale, spinach, raisins, Cheddar, chicken, bacon, roasted nuts, 1 teaspoon salt and 1/2 teaspoon pepper. Before serving, grate the apple and add it. Pour the dressing over the salad and toss gently. Put the eggs on top, toss lightly, and serve.