Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.
Step 2: In a large pot or Dutch oven, heat olive oil over medium heat. Add onion and cook until softened, about 5-7 minutes. Add minced garlic and cook for another minute until fragrant.
Step 3: Add diced potatoes, carrots, parsnips, and broccoli florets to the pot. Pour in about 1/2 cup of vegetable broth, cover, and simmer for 10-15 minutes, or until vegetables are tender-crisp. Drain any excess liquid and set aside.
Step 4: In a separate saucepan, melt butter over medium heat. Whisk in flour and cook for 1-2 minutes, stirring constantly, to create a roux.
Step 5: Gradually whisk in the warm milk and remaining vegetable broth until the sauce is smooth and thickened. Reduce heat to low. Stir in the softened cream cheese until fully melted and smooth.
Step 6: Remove from heat and stir in thyme, nutmeg, salt, and pepper. Add 3/4 cup of Gruyère and 3/4 cup of cheddar cheese to the sauce, stirring until melted and well combined.
Step 7: Gently fold the cooked vegetables into the creamy cheese sauce until evenly coated. Pour the mixture into the prepared baking dish.
Step 8: In a small bowl, combine the remaining 1/4 cup Gruyère, 1/4 cup cheddar, and panko breadcrumbs. Sprinkle this mixture evenly over the top of the casserole.
Step 9: Bake for 25-30 minutes, or until the casserole is bubbling around the edges and the topping is golden brown and crispy. If the topping browns too quickly, you can loosely cover it with foil.
Step 10: Let the casserole rest for 5-10 minutes before serving. Garnish with fresh parsley, if desired.