Pat the chicken dry and place it in an oval slow cooker.
In a bowl, whisk together the oil, brown sugar, chili powder, smoked paprika, cumin, garlic powder, onion powder, mustard, salt, and pepper. It should form a paste.
Working gently, use your hand to loosen the chicken's skin from the meat, starting at the top of the breast and working your way down and over the thighs (I wear a rubber glove). Rub half of the mixture underneath the skin so it's directly touching the meat. Rub the rest of it all over the outsides of the chicken, both front and back.
Turn the chicken BREAST SIDE DOWN (this allows the flavorful drippings from the thighs to keep the breast moist). Cover and slow cook on LOW heat until the breast registers at least 160°F and the thighs register at least 175°F, about 3 to 4 hours depending upon the size of your chicken and your slow cooker model (mine finishes in 3 hours, 20 minutes). You do not need to add any liquid to the slow cooker; the chicken will release plenty of moisture.
To brown the skin: Using two forks (be careful, the chicken is hot!), carefully remove the chicken from the slow cooker and place it breast-side up on a parchment-lined baking sheet. Place on the center rack and broil for a few minutes, watching carefully the entire time and rotating the baking sheet as needed, until the skin is as dark as you like (I broil for about 5 minutes or so).
Transfer the chicken to a cutting board. Let rest 10 minutes; it will be falling apart! Slice or pull and use as desired.
Notes
TO STORE: Refrigerate leftovers in an airtight container for up to 4 days.
TO REHEAT: Warm gently in the microwave or oven.
TO FREEZE: Freeze shredded chicken for up to 3 months.
Breast side down cooking method adapted from ATK; flavors inspired by Costco rotisserie chicken