Combine chicken breast, 1/2 of the salsa verde, chicken broth, onion, chili powder, cumin, cayenne, salt, and pepper in a slow cooker. Cook until chicken is cooked through on Low for 6 hours or High for 3 hours.
Remove chicken from sauce and shred in a bowl with 2 forks. Return to the slow cooker and mix to combine with sauce.
Place a large skillet over medium heat; add olive oil and heat until it starts to shimmer. Add tortilla triangles to the hot oil and cook, flipping every minute, until golden, about 8 to 10 minutes.
Add shredded chicken, slow cooker sauce, and remaining salsa verde to tortillas. Mix and heat until everything is warmed through.
Meanwhile, stir sour cream and enough milk together so it can be drizzled.
Remove tortilla mixture from the heat. Sprinkle cotija cheese on top and garnish with cilantro; drizzle with thinned sour cream mixture.