Position racks in the top and bottom thirds of the oven and heat the oven to 450°F. Butter a 9x13-inch (or similar) baking dish.
In a large bowl, toss the white mushrooms with 2 Tbs. of the oil, 1/2 tsp. salt, and 1/4 tsp. pepper. Spread the white mushrooms evenly on a large rimmed baking sheet. Toss the “wild” mushrooms with the remaining 2 Tbs. oil, 1/2 tsp. salt, and 1/4 tsp. pepper in the bowl and spread on a second large rimmed baking sheet. Roast, stirring the mushrooms and rotating the baking sheets once, until golden-brown and any liquid has evaporated, about 25 minutes. Reduce the oven temperature to 350°F.
Meanwhile, melt 2 Tbs. of the butter and pour into a medium bowl. Add the breadcrumbs and fontina and toss well.
In a 12-inch skillet, melt the remaining 2 Tbs. butter over medium heat. Add the leeks and 1 tsp. salt and cook, stirring often with a wooden spatula, until golden, 7 to 10 minutes. Add the vermouth and boil, stirring and scraping up any browned bits, until reduced by half, 1 to 2 minutes. Stir in the heavy cream, sage sprigs, 1/2 tsp. salt, and 1/4 tsp. pepper and bring just to a boil over high heat. Lower the heat to medium low and simmer for 5 minutes to infuse. Remove the pan from the heat and discard the sage sprigs.
Add the mushrooms and chopped sage to the leek mixture and toss to coat. Transfer to the prepared dish. Top with the breadcrumb mixture and bake in the top third of the oven until the breadcrumbs are golden-brown, about 30 minutes.
Let rest for 15 minutes before serving.