Preheat the oven to 375 degrees F; butter a shallow 3-quart baking dish. Melt 2 tablespoons butter in a large nonstick skillet over medium-high heat. Add the mushrooms, season with salt and cook, stirring occasionally, until tender and lightly browned, 8 to 10 minutes. Transfer the mushrooms to a plate and wipe out the skillet.
Melt 6 tablespoons butter in the same skillet over medium heat. Add the leeks; cook, stirring occasionally, until tender, about 6 minutes. Add the mushrooms, thyme, rosemary, broth, 1 teaspoon salt and a few grinds of pepper. Bring to a simmer, then remove from the heat.
Whisk the eggs and heavy cream in a large bowl. Add the herbs, bread cubes and mushroom-broth mixture and stir until combined. Transfer the stuffing to the prepared baking dish. Cut the remaining 2 tablespoons butter into small cubes and scatter evenly over the stuffing. Cover with foil and bake 30 minutes, then uncover and bake until golden, 20 to 30 more minutes.