Trim a narrow strip off one end of each pita and open the pitas with your fingers.
In a large nonstick skillet, heat the olive oil and butter over medium-high heat until the butter stops foaming. Add the cremini mushrooms and cook, stirring occasionally, until they begin to brown, about 5 minutes. Push the cremini to the side of the skillet and add the white mushrooms. Cook, stirring occasionally, until the white mushrooms begin to brown, about 5 minutes.
Stir in the arugula and heat until just wilted; season the mixture with salt and pepper to taste and remove from the heat. Mix in the chicken and spoon the filling into the pitas. Serve warm.