Preheat the oven to 375 degrees F. Butter an 8-by-8-inch baking dish and set aside.
For the casserole: Fill a large pot with water. Add the garlic and a small handful of salt. Bring to a boil and add the potatoes. Cook until fork tender, 6 to 8 minutes. Remove the potatoes and garlic with a slotted spoon to a sieve to let drain. To the same water, add the green beans. Bring back to a boil if needed, then simmer until the green beans are bright and tender, 6 to 8 minutes. Remove to a sieve.
In a saute pan on medium heat, add the olive oil, onions, serrano chile and thyme. Season with a pinch of salt, a grind or 2 of pepper and paprika. Cook until the onions are tender, and then add the mushrooms and Worcestershire sauce, and season again with salt and pepper. Cook until the mushrooms release some of their moisture and become tender, about 10 minutes. Remove the thyme sprigs and slowly stir in the heavy cream. Raise the heat until things begin to bubble. Cook at this temperature, while stirring, until the cream thickens a bit, 6 to 8 minutes. Once thickened, turn off the heat and stir in the crema.
In one large bowl, add the potatoes. In another large bowl, add the green beans. Add the jalapenos to the potatoes. Then, add three-quarters of the mushroom mixture to the potatoes and the rest to the green beans. Divide the cheese the same way into both bowls. Mash the potatoes until combined. Press into the bottom of the prepared baking dish. Stir the green beans until combined, and spread evenly over the top.
For the topping: Rinse the potato peels and dry as much as possible with paper towels. In a bowl, add the potato strips, sage, thyme, paprika and olive oil. Season with a pinch of salt and a grind or 2 of pepper. Toss and sprinkle evenly over the top of the green beans. Bake until the potatoes brown and begin to crisp on the edges, about 30 minutes.
To prepare in advance, build casserole, but leave off the topping and refrigerate. Refrigerate the reserved potato peels already cut into strips in a container with water. Before baking, rest the casserole at room temperature for about 1 hour, and then make the topping and bake as instructed.