Rub the chicken all over with salt, pepper, paprika, and thyme, then set aside to marinate briefly.
Heat olive oil in a skillet over medium-high heat, then sear the chicken until golden brown on all sides—this develops flavor and a crispy skin look.
Place the sliced onion, carrots, and celery into the bottom of the slow cooker, creating a bed for the chicken.
Transfer the seared chicken on top of the vegetables, then pour chicken broth over everything. Add minced garlic on top of the chicken for added aroma.
Cover with the lid and cook on low for about 6 hours, until the chicken is fall-off-the-bone tender and the vegetables are soft and flavorful.
Once cooking is complete, carefully remove the chicken and set it on a baking sheet. Broil in the oven for 3-4 minutes until the skin crisps up and turns a golden brown.
Meanwhile, spoon the caramelized vegetables and cooking juices from the slow cooker into a serving dish; taste and adjust seasoning as needed.
Slice the crispy-skinned chicken and arrange it on a platter alongside the vegetables. Pour some of the juices over the top for extra flavor.
Serve hot, enjoying the tender meat with crispy skin and the savory, caramelized vegetables.