Preheat the oven to 350°F. Remove a few large pieces of skin from the rotisserie chicken and slice each piece into about 1/4-inch-wide strips. Arrange the strips on a baking sheet lined with foil or parchment and lightly brush the skin pieces with olive oil. Bake 12–15 minutes, until the skins are crispy. Remove from the oven and let cool, then break or crumble into small pieces; set aside.
While the skins bake, remove the meat from the rest of the rotisserie chicken (about 1/2 rotisserie chicken) and shred it into bite-sized pieces. Place the shredded chicken in a large bowl.
Add 3 tablespoons mayonnaise, 1 tablespoon chopped fresh parsley, 1 tablespoon sweet pickle relish, 2 teaspoons stone-ground or Dijon mustard, and 2 teaspoons small capers to the shredded chicken. Toss until everything is evenly coated. Taste and season with salt and pepper to your preference.
Brush both sides of the 6 slices of sourdough bread lightly with olive oil. Toast the bread by grilling or cooking in a cast-iron skillet over medium heat until each slice is crispy and browned in spots (or use a toaster if preferred).
Divide the chicken salad evenly among the toasted sourdough slices, piling it on top. Sprinkle the crumbled crispy chicken skins over each tartine and finish with microgreens, if using. Serve immediately.