Preheat oven to 375°F. Oil four 8-ounce ramekins; place on a rimmed baking sheet.
Heat oil in a large skillet over medium heat. Add onion and cook, stirring occasionally, until tender, 5 to 7 minutes. Add carrots and season with salt and pepper. Cook, stirring occasionally, until carrots are just tender, 8 to 10 minutes. Stir in garlic and thyme. Cook until fragrant, 30 seconds.
Sprinkle flour over vegetables and cook, stirring, 1 minute. Slowly stir in milk. Simmer until slightly thickened, 2 to 3 minutes. Fold in parsley. Transfer to prepared ramekins, dividing evenly.
Cut 4 4 1/2-inch rounds from puff pastry; place 1 round over each ramekin. Brush with egg. Bake until puffed and golden brown, 20 to 25 minutes.