1. Position a rack in the lower third of the oven and preheat the oven to 375°F.

2. Place the carrots, celery, onion, and garlic in a large Dutch oven and season with salt and pepper. Bake the vegetables, uncovered, stirring them once or twice, until they are tender and browned, about 1 hour.

3. Transfer the Dutch oven to the stove top. Add the stock and 3 cups of warm water and bring to a boil over high heat. Add the bay leaves and turkey giblets and neck. Reduce the heat to low and let simmer until the liquid is reduced to about 3 cups, 1 to 1½ hours.

4. Strain the stock through a sieve into a medium-size saucepan, pressing hard on the solids to extract all the liquid. Discard the solids. If you are planning on adding the turkey pan drippings, refrigerate the reduced stock, covered, until you are ready to proceed.

5. Place 1 cup of the strained stock in a medium-size bowl and add the half-and-half and flour. Whisk until smooth.

6. Place the saucepan with the remaining stock over high heat and bring to a boil (see Notes). Stir in the flour mixture. Reduce the heat to low and let simmer, whisking frequently, until the gravy thickens and the raw taste of the flour cooks off, about 15 minutes. If the gravy is too thick, add more half-and-half or water. If it is not thick enough, add ½ to 1 tablespoon more flour and let the gravy cook for 3 to 5 minutes longer. Taste for seasoning, adding more salt and/or pepper, as needed.

7. If you are not planning on serving the gravy immediately, transfer it to a clean container and let it cool. The gravy may be refrigerated, covered, for up to 2 days. Reheat over medium-low heat.