Bring a medium pot filled halfway with water to a rolling boil over high heat. Add ¼ cup of salt and stir until all the salt is dissolved. The water should taste similar to ocean water.
French the beans, removing the strings on both sides with a paring knife. Add the green beans to the water and cook until tender, about 3 minutes.
Drain the green beans and add to a large bowl filled with cold water and ice cubes to stop the cooking process. When the green beans are cool remove them from the ice and drain again. Set aside until ready to use.
Heat oven to 375ºF (190ºC).
Heat a large sauté pan over medium heat, add the olive oil and heat through. Add the mushrooms and sauté, stirring occasionally, until golden brown and reduced in size, about 6 minutes. Season with salt then add the butter and garlic, cooking until the garlic is fragrant, about 30 seconds.
Stir in the flour and let cook, continuing to whisk frequently until the mixture is pale brown. Add the milk, stirring to combine. Cook, stirring occasionally, until the mixture has thickened, about 5 minutes. Season to taste with salt and pepper.
Add the green beans to a large baking dish and pour the cream mixture over the top, tossing to combine. Top with the parmesan and bake until the top is bubbling, about 20 minutes.