Preheat the oven to 350°F. Remove the skin from the 1/2 rotisserie chicken and slice the skin into about 1/4-inch–wide strips. Arrange the strips on a parchment- or foil-lined baking sheet.
Bake the skin strips 12–15 minutes, or until crispy. Remove from the oven and let cool on the sheet; when cool, crumble the crispy skins into small pieces and set aside.
While the skins bake, shred the meat from the 1/2 rotisserie chicken into bite-sized pieces and place the shredded meat in a mixing bowl.
Add 3 tablespoons mayonnaise, 1 tablespoon chopped fresh parsley, 1 tablespoon sweet pickle relish, 2 teaspoons mustard, and 2 teaspoons capers to the shredded chicken. Toss to coat evenly. Taste and season with salt and pepper to your preference.
Toast the 6 slices of sourdough bread. After toasting, brush each slice lightly with olive oil.
Divide the chicken salad evenly among the 6 toasted sourdough slices, piling as desired.
Sprinkle the crumbled crispy chicken skins over each tartine and finish with microgreens, if using.