1. Preheat oven to 400 degrees F. On a lined baking sheet, lay the game hens flat, skin-side up. Season with salt, black pepper, and dried thyme.
In a large cast iron skillet, heat 2 tablespoons of unsalted butter over medium heat. When the butter is melted, using tongs, place the game hens skin side down into the skillet. Gently press down on the hens with tongs and cook until skin is golden brown and crispy, about 8-10 minutes.
2. Flip the game hens and cook for 68 more minutes. Turn off the heat and transfer the game hens back to the lined baking sheet, skin side up, and roast for 20-25 minutes, until cooked through.
3. In the same skillet over medium-‐low heat, add the shallots and cook for 10-15 minutes, until translucent. Add the garlic and fresh thyme to the pan and cook for 2 minutes, until fragrant. Deglaze the pan with the lemon juice and white wine and whisk until slightly reduced, about 6-7 minutes. Whisk in remaining butter.
4. When the game hens are done, cover with tinfoil and let rest for 10-15 minutes.
5. Spoon the warm sauce over the Cornish game hens before serving, garnish with more fresh thyme sprigs and serve with fresh lemon wedges.