In a medium bowl, whisk together broth, wine, tomato paste, and flour until evenly combined and smooth. Stir in dried mushrooms and set aside to reconstitute, at least 30 minutes. Strain mushrooms before using, reserving liquid.
Place potatoes in a large pot and cover with heavily salted water by 2 inches. Bring potatoes to a boil and cook until fork tender, about 20 to 30 minutes.
Heat the oven on low broil and place a rack in the upper third. Meanwhile, melt 1 tablespoon of the butter in a 3 to 4-quart Dutch oven (or oven-ready saucepan) over medium-high heat. When it foams, add half the mushrooms and cook, stirring rarely, until mushrooms are browned, about 5 minutes. Remove mushrooms from pan, season well with salt and freshly ground black pepper, and set aside. Repeat to cook off remaining mushrooms.
Return pan to stove over medium heat and add 1 tablespoon of the butter, onion, celery, and garlic, and cook until softened and golden, about 2 minutes. Add celery root, carrot, parsnip, and herbs, and season well with salt and freshly ground black pepper. Cook until browned and softened, about 6 minutes.
Add wine mixture to pan and deglaze by stirring and scraping up any browned bits. Let cook until simmering and slightly thickened, about 3 minutes. Stir in reserved mushrooms and any juices that have accumulated and simmer until slightly thickened, about 8 minutes. Remove from heat and reserve in pan.
When potatoes are ready, drain well. Return to pan and mash until uniformly smooth. Fold in remaining 2 tablespoons butter and milk, and season well with salt and freshly ground black pepper. If necessary, keep warm over low heat.
Dot potatoes over vegetable mixture and spread to edges of pan to cover completely. Rough up the surface of the potatoes so there are bits that will get browned and crunchy. Broil until top is golden, about 15 to 20 minutes. Serve.