1. Preheat the oven to 350°. Butter a 10-inch glass pie plate and sprinkle it lightly with Parmesan cheese.
2. In a large saucepan, steam the carrots and garlic until very tender, about 25 minutes. Transfer the carrots and garlic to a food processor and puree until smooth. Pour in the heavy cream and milk and process to blend. Add the eggs, 2 tablespoons of the Parmesan cheese, 1 teaspoon of salt and 1/4 teaspoon of pepper and process the mixture until thoroughly blended.
3. Pour the flan mixture into the prepared pie plate. Set the pie plate in a shallow roasting pan and add enough hot water to the pan to reach halfway up the side of the pie plate. Bake for 40 minutes, or until the flan is just set.
4. Meanwhile, in a small bowl, toss the bread crumbs with the melted butter, thyme and the remaining 2 tablespoons of Parmesan cheese. Season the thyme crumbs with salt and pepper.
5. Remove the flan from the water bath. Preheat the broiler. Sprinkle the flan with the thyme crumbs and broil for 30 seconds, or until golden brown. Cut into 6 wedges and serve warm.
Make Ahead: The flan can be assembled and refrigerated for 4 hours. Bring to room temperature before baking.