Coarsely grind caraway seeds with salt with a mortar and pestle. Mash the garlic into a paste with the caraway. In a large bowl, whisk together the garlic paste with the honey, lemon juice, and olive oil. Season with paprika, cumin, ginger, cinnamon, cayenne, and black pepper; mix well.
Lightly oil the inside of a heavy, ovenproof Dutch oven. Add the zucchini, turnips, red pepper, yellow squash, and onions. Toss with 1/2 of the spice paste. Stir in tomatoes and chicken broth. Toss the Cornish hens with the remaining spice paste, then arrange them in the Dutch oven, so that the cut side is facing down.
Cover the Dutch oven, and bake in the center of the preheated oven for 1 hour and 15 minutes. If a crunchy skin is desired on the Cornish hens, uncover during the last 15 minutes of cooking. Skim the fat off of the broth.
Serve hens on top of vegetables, spoon some of the broth over, and sprinkle with the parsley, cilantro, and mint.
