Instead of the usual roast chicken, why not give game hen the oven treatment? It cooks way faster, and the pan juices make a delicious sauce when combined with Dijon, lemon juice, capers, and fresh tarragon.
Preheat oven to 425 degrees with racks in top and bottom. Place hen, breast-side down, on a work surface. Using kitchen shears, cut along both sides of backbone to remove. Flip over; split lengthwise with a heavy knife. Pat dry; season with salt and pepper. On a rimmed baking sheet, toss vegetables with 1 tablespoon oil; season with salt and pepper. Transfer sheet to bottom rack of oven.
In an ovenproof skillet, heat remaining oil over medium-high. Add hen, skin-side down; cook until browned, about 3 minutes. Flip; transfer skillet to top rack of oven. Roast, stirring vegetables once, until golden, and a thermometer inserted into thickest part of hen thigh reads 165 degrees, 18 to 20 minutes for hen, and 25 to 28 minutes for vegetables. Transfer hen and vegetables to a platter. Return skillet to medium heat; whisk in mustard, lemon juice, 1 tablespoon water, and capers. Cook until reduced slightly, about 1 minute. Stir in tarragon; spoon over vegetables. Serve, with more mustard.
If you can't find celery root, parsnips are another great match for the carrots and game hen.