Preheat the oven to 425 degrees F; line a baking sheet with foil. Cut the hens in half with kitchen shears, cutting along one side of the breastbone; season with the coriander, 1 teaspoon salt and a few grinds of pepper. Heat 1 tablespoon butter in a large skillet over medium-high heat. Add the hens skin-side down; cook until golden, 4 minutes. Flip; cook until slightly golden, 2 more minutes. Transfer to the prepared baking sheet skin-side up. Bake until cooked through, 25 minutes.
Meanwhile, add the broth, thyme sprigs and orange zest and juice to the skillet. Bring to a boil, stirring. Reduce the heat to low; cook until thickened, 15 minutes. Season with salt and pepper. Set aside.
Heat 2 tablespoons butter in a medium skillet over medium-high heat. Add the orzo, shallot and chopped thyme. Cook, stirring occasionally, until the orzo is lightly toasted, 5 minutes. Add 2 cups water and 1/2 teaspoon salt; bring to a boil. Reduce the heat to medium low, cover and cook until the orzo is al dente, 12 minutes. Fluff the orzo; stir in the remaining 1 tablespoon butter, the carrots and parsley. Season with salt and pepper. Serve with the hens and the reduced sauce.