1. Place dried morel mushrooms in a large heatproof bowl; add 4 cups hot water. Place a plate on top of mushrooms to keep submerged. Let stand until mushrooms are plump, about 30 minutes. Lift mushrooms from liquid, and squeeze gently to remove excess liquid. Reserve liquid in bowl; finely chop mushrooms.
2. Melt 1/4 cup butter in a large (12- to 14-inch) skillet over medium. Add onion and carrot; cook, stirring often, until carrot is tender, about 8 minutes. Add cremini mushrooms, celery, garlic, and chopped rehydrated morels; cook, stirring occasionally, until cremini mushrooms have released their liquid, about 5 minutes. Sprinkle mixture with pepper and 1 teaspoon salt; cook, stirring often and scraping bottom of skillet, until liquid has evaporated and vegetables start to caramelize, about 12 minutes. Add wine and thyme sprigs; stir to release browned bits from skillet. Cook, undisturbed, until wine has completely evaporated, about 2 minutes. Carefully pour reserved mushroom soaking liquid into skillet, leaving any sediment in bowl. Add broth to skillet; bring to a vigorous simmer over medium. Reduce heat, and simmer, stirring occasionally, until mixture has thickened and reduced to about 3 cups, 40 to 45 minutes. Discard thyme sprigs. Cover and keep warm over low until ready to use.
3. Bring 4 quarts water to a boil in a large pot over high. Add remaining 1/4 cup salt, and return to a boil. Add bucatini; cook, stirring occasionally, until pasta is al dente, 7 to 9 minutes, adding dandelion greens during last minute of cooking. Meanwhile, return ragù in skillet to a simmer over medium-high.
4. Using a spider or fine wire-mesh strainer, lift pasta and greens from cooking liquid, and add to ragù in skillet. Cook over medium-high, tossing to coat pasta with sauce. Add pasta cooking liquid as needed, 1/4 cup at time, until sauce is creamy and clings to pasta, about 4 minutes. Stir in mustard and remaining 1/4 cup butter until combined. Season with salt and pepper to taste. Fold in tarragon.
5. Divide pasta evenly among bowls; sprinkle with Parmesan. Garnish with tarragon leaves and additional pepper. Serve immediately.