Preheat the oven to 375°F (190°C).

Heat olive oil in a large skillet over medium heat. Add the onion, carrots, celery, and garlic. Cook until the vegetables are softened, about 5-7 minutes. Stir in the shredded chicken and frozen peas.

In a separate saucepan, melt the butter over medium heat. Stir in the flour and cook for 1-2 minutes. Gradually whisk in the chicken broth and cook until thickened, about 5 minutes.

Stir in the heavy cream, thyme, rosemary, salt, and pepper. Pour the sauce over the chicken and vegetable mixture, stirring to combine.

In a lightly greased baking dish, layer 5 sheets of phyllo dough, brushing each sheet with melted butter before adding the next.

Spread the chicken mixture evenly over the phyllo layers. Top with the remaining 5 sheets of phyllo dough, again brushing each with melted butter.

Place the dish in the preheated oven and bake for 35-45 minutes, or until the phyllo is golden and crisp.

Let the pie cool for a few minutes before serving.