Melt butter in a large skillet over medium-high heat. Add shallot and garlic; cook and stir for 30 seconds. Add mushrooms, 1/2 teaspoon salt, and white pepper. Reduce heat to medium; cook, stirring often, until moisture has evaporated and mushrooms start to brown, about 12 minutes.
Stir walnuts and white wine into the skillet. Cook, scraping browned bits off the bottom of the skillet, until wine has evaporated, 3 to 5 minutes. Remove from heat and let cool, 10 to 15 minutes.
Preheat oven to 400 degrees F (200 degrees C).
Season Cornish hens inside and out with remaining 1 teaspoon salt, paprika, onion powder, garlic powder, and black pepper. Open cavities gently with your fingers; stuff with mushroom mixture. Pull skin back in place to cover stuffing. Place hens breast side-up in a roasting pan.
Roast in the preheated oven, basting once or twice with juices from the pan, until an instant-read thermometer inserted into the thickest part of the thigh registers 180 degrees F (82 degrees C), about 45 minutes. Garnish with thyme sprigs.