In a large skillet or Dutch oven, cook lardons over medium-low heat to render the bacon fat. When the bacon pieces are crisp, remove with a slotted spoon (leaving bacon rendering in pan) and set aside.
Bring a large pot of salted water to a boil to a boil.
Add the diced yellow onion to the bacon pan and turn the heat to medium. Sauté onions, stirring often, for about 3 minutes, until just opaque. Add butter and mushrooms to the pan and cook about 8 more minutes, stirring frequently. Mushrooms will give off quite a bit of liquid, which will then cook down and thicken.
Add the flour to the pan and cook, stirring almost continuously, for about 3 minutes, being mindful not to burn the mixture (reduce heat as necessary).
Whisk in the milk, and stir until smooth and thickened, about 8-10 minutes. Avoid scorching by stirring frequently and scraping the bottom of the pan with a wooden or silicone spoon.
Wash the Swiss chard leaves in cool water to remove any grit. Trim to just below the leaves and cut them into large ribbons across the stem. Add ribbons to the boiling water and blanch for 2 minutes. Remove ribbons from boiling water and shock them in an ice cold water bath. So as to keep the water boiling, do this in batches. Drain in colander.
Allow the greens to cool. Squeeze greens to remove excess water.
Add the Swiss chard and bacon lardons to sauce mixture and mix evenly.
Pour mixture into a 5 to 6 cup gratin baking dish. Sprinkle breadcrumbs over gratin to cover. Bake for about 50 minutes, minutes turning halfway through the cooking time, until gratin is browned and bubbly. Let the gratin rest for 20 minutes before serving.