Preheat the oven to 400 degrees. Place one oven rack about 6 inches from the top and another in the center of the oven. Place the green beans on a large rimmed baking sheet and drizzle with oil. Sprinkle with salt and pepper and toss with your hands to coat. Roast the beans on the center rack for 8 minutes.
While the beans are roasting, melt the butter in a large skillet. Add the flour and whisk it into a paste. Cook over medium heat for about a minute. Whisk in the water and let the sauce simmer over low heat for 3-5 minutes, stirring occasionally, until it has thickened.
Remove from the heat and whisk or stir in 1/2 cup of the grated Parmesan cheese and all of the Monterrey Jack cheese, continue to stir until the sauce is smooth. Stir in the salt and pepper and adjust as needed. Cover with a lid and let stand 5 minutes.
Remove the green beans from the oven and increase heat to broil. Toss the beans lightly with tongs (or stir with a large spatula) and pour sauce over them. Sprinkle with the remaining 1/2 cup of Parmesan cheese. Place the green beans back in the oven on the upper rack under the broiler. Broil for 3-4 minutes, watching closely, just until the sauce is bubbling and the beans and sauce are beginning to brown. Serve hot. Enjoy!
Green Bean Gratin https://barefeetinthekitchen.com/green-bean-gratin-recipe/