Steam the frozen California mix vegetables until they are just tender. Drain any excess water.
In a saucepan, melt butter over medium heat. Add flour and stir continuously to form a roux. Cook for 1-2 minutes, being careful not to brown the mixture. Gradually whisk in the milk to the roux, ensuring there are no lumps. Cook the sauce until it thickens, stirring constantly. Add 1 cup of the shredded cheddar cheese to the sauce and stir until the cheese is fully melted. Season with salt and pepper to taste.
Place the cooked California mix vegetables in a baking dish. Pour the cheese sauce over the vegetables, ensuring they are well coated. Add the remaining cup of cheese on top of the cheese sauce.
Bake in the preheated oven for about 20-25 minutes or until the top is golden brown, and the gratin is bubbly around the edges.
Remove from the oven and let it rest for a few minutes. Serve hot.
