Bring a large (5-7 quart) pot of water to a boil, and have a large bowl filled with ice water ready in the freezer.
While the water is boiling, prep each of the following and set each aside separately. For the green beans, snap the stems off and slice longer beans in half so that you end up with pieces that are about 2-3" long. For the cremini mushrooms, trim off the stem ends and slice the mushrooms 1/4" thick. Mince the garlic.
When the water is boiling, remove two cups of the water to a heat-proof bowl or mixing cup. Stir in the dried porcini and let steep for at least 20 minutes.
Preheat the oven to 400°F and place a rack in the middle of the oven. Have a 9"x13" casserole dish standing by on the counter.
Return to the large pot of boiling water and add 2 tsp salt. Take the bowl of ice water out of the freezer. Add the green beans to the boiling water and cover so that the water comes back to a boil more quickly. Blanch the beans for about 2-4 minutes until bright green and crisp-tender, then remove with a slotted spoon to the ice water to stop the cooking, then drain in a colander and set aside to dry.
Line a fine-mesh strainer with a paper towel, then strain and reserve the liquid from the porcini, pressing the mushrooms to release as much liquid as possible. Discard the porcini.
Heat the olive oil in a large saute pan over medium high heat. Add the reserved sliced onions as well as the garlic to the pan and toss/stir to coat with the oil. Stir frequently to prevent the garlic from burning. When the onions start to turn translucent, add the sliced cremini mushrooms and toss to distribute. Continue to saute as the mushrooms release their juice and the mushrooms and onions begin to brown.
Add the reserved porcini broth to the pan and stir to distribute. Saute until the porcini broth has been fully absorbed and the mushrooms, onions, and garlic are golden brown. Remove from heat and add the mixture to the empty casserole dish.
Separate 6 tablespoons of the flour and reserve the rest, keeping the 6 tablespoons nearby for the béchamel sauce.
Heat the butter in a medium sauce pan over medium-high heat. When the butter is melted, add the 6 tablespoons of flour and whisk to combine, stirring constantly. When the flour starts to turn light brown, add the milk in a steady stream as you whisk it in. Keep stirring until the béchamel has thickened. Turn off heat. If you're opting to use Asiago, add it now and stir to melt. Add salt and pepper to taste.
Add the now-dry green beans to the onion-garlic-mushroom mixture in the casserole dish and pour the béchamel in, stirring gently to coat everything. Bake uncovered for 20-30 minutes until the mixture is bubbling and starting to brown.