Pre-heat oven to 400 degrees.
Heat olive oil in a large saucepan over medium heat. Add carrots, onion, and thyme, salt, and pepper. Cook until carrots are tender, but still crisp, 8-10 minutes. Add garlic and cook for one more minute. Add flour, and cook while stirring for 1 minute. Slowly add milk, stirring until smooth. Continue to cook, stirring occasionally, until mixture comes to a simmer and thickens.
Remove saucepan from heat and stir in peas, lemon juice, shredded chicken, and ¼ cup chopped parsley. Pour filling into a 9-inch deep-dish pie plate or square baking dish.
Stack phyllo on a work surface. Using a paring knife, cut out an 11-inch circle or square from the stack (depending on if your baking dish is round or square) and discard trimmings. Brush 1 sheet of phyllo with melted butter and sprinkle with remaining parsley leaves. Repeat with remaining phyllo, parsley, and butter; do not sprinkle parsley on final layer. Drape crust over top of dish; tuck in edges. Cut four slits in top of phyllo for vents.
Bake pot pie until golden and bubbling, 20 to 25 minutes. Let cool 15 minutes before serving.