Heat the coconut oil in a skillet over medium heat, and cook the plantain pieces just until tender but still firm. Remove from heat, and drain on paper towels. Stand the plantain pieces on end, and with a small dish or the bottom of a cup, gently flatten the plantains. Return to oil, and continue cooking until golden brown. Remove from heat, and drain on paper towels.

Place the mushrooms in the skillet with remaining coconut oil over medium heat, and cook until tender.

Stuff pita bread halves with equal amounts of the plantain, mushrooms, avocado, and arugula to serve.