Place the chicken breasts in the slow cooker and pour the 16-ounce jar of green chile salsa over them, turning to coat so the chicken is mostly submerged.
Cover and cook on low for 4 to 6 hours, until the chicken is cooked through and reaches an internal temperature of 165°F (74°C).
Transfer the chicken to a cutting board and shred it with two forks, then return the shredded chicken to the slow cooker and stir to coat with the salsa; keep warm until serving.
While the chicken cooks, combine the thinly sliced red cabbage and grated carrot in a mixing bowl; drizzle with 3 to 4 tablespoons fresh lime juice and 1 teaspoon sea salt.
Use your hands to massage the lime and salt into the cabbage and carrot until the cabbage softens slightly, then taste and adjust lime juice or salt as needed.
To assemble bowls, divide 5 to 6 cups cooked brown rice among 4 bowls, top with shredded chicken, a portion of the cabbage-carrot slaw, sliced avocado, and chopped cilantro.
Serve with lime wedges and additional toppings if desired.