Preheat the oven to 450 degrees . Slice the potatoes lengthwise 1/3 inch thick, then stack and thinly slice to shape into french fries. Transfer to a large bowl and toss with 2 tablespoons olive oil. Arrange the potatoes in an even layer on a parchment-paper-lined rimmed baking sheet, reserving the bowl; season with salt and pepper. Bake until deep golden, about 35 minutes.
Meanwhile, in a large skillet, heat the remaining 1 tablespoon olive oil over medium-high heat. Add the mushrooms and cook, stirring occasionally, until browned, about 5 minutes; transfer to the reserved bowl. Add the butter and onion to the pan and cook until the onion begins to brown, about 3 minutes; lower the heat to medium-low and cook until softened, about 5 minutes. Stir in the flour and cook for 1 minute. Stir in the chicken broth, tomato paste and cooked mushrooms and simmer until thickened, about 3 minutes; season with salt and pepper. Remove from the heat and cover to keep warm.
Sprinkle two-thirds of the cheese on top of the fries and return to the oven to melt, about 1 minute; reheat the mushroom gravy, if needed. Using a metal spatula, transfer half of the fries to a platter and spoon half of the gravy on top, then repeat with the remaining fries and gravy; sprinkle the remaining cheese on top.
