Season both sides of chicken breast with 1 tablespoon berbere spice and 1 teaspoon kosher salt.

Melt butter in a skillet over medium-high heat until it just starts to turn brown. Add chicken breast to the skillet and cook 2 to 3 minutes per side; transfer to a warm dish.

Pour coconut milk, chicken broth, lime juice, and tomato paste into the skillet; bring to a simmer while scraping browned bits of food off the bottom of the skillet with a wooden spoon. Add 1/2 teaspoon berbere spice and pinch of salt.

Return chicken breast to the skillet; reduce heat to medium-low. Cook until chicken is cooked through and sauce starts to reduce, 4 to 5 minutes, basting chicken with pan juices as it cooks. An instant-read thermometer inserted into center should read at least 165 degrees F (74 degrees C).

Sprinkle chicken with chopped cilantro; drizzle with sauce.