Combine the onion, carrots, celery, garlic, and olive oil in a large skillet. Cook over medium heat for 5-7 minutes, or until the veggies are softened.
Stir in the cooked chicken and frozen peas.
Melt the butter over medium heat in a separate medium saucepan.
Whisk in the flour and cook for 1-2 minutes.
Continue whisking as you add the broth. Cook until the mixture has thickened. This should take no more than 5 minutes.
Stir in the heavy cream, thyme, rosemary, salt, and pepper.
Pour this sauce over the chicken mixture and stir to combine.
Assemble
Grease a baking dish with butter.
Layer 5 sheets of phyllo dough into the baking dish and then brush each of them with melted butter.
Add the chicken filling mixture on top of the phyllo layers.
Top with 5 more sheets of the dough and then coat those with melted butter as well.
Bake for 35-45 minutes or until the phyllo is flaky and golden brown.
Allow to cool for several minutes before serving. This allows the filling to thicken just a bit, which helps keep it together when portioning the pie.