Rinse and pat the chicken dry. On a stable surface, cut the chicken with a sharp knife into 8 pieces, plus the backbone.
Season all the pieces with 1 teaspoon each salt and pepper. Cover and refrigerate all but the backbone.
In a large saucepan or small stockpot over high heat, add 1 tablespoon of garlic oil and, when hot, add in the seasoned chicken backbone. Brown for 4 to 5 minutes, then add in the reserved garlic cloves, thyme, and the cut vegetables. Turn the backbone over and continue to cook until well browned, an additional 5 minutes. (Guy's tip: for easy removal when making gravy later, tie the garlic cloves in cheesecloth before adding to the pot).
Deglaze the pan with a little of the stock to loosen the fond from the bottom of the pan, and add the remaining stock and the water. Bring to a strong simmer and let cook for 45 minutes. Strain the solids but reserve the garlic cloves in a bowl. Let sit and skim the fat from the top. Set aside. You should have about 3 cups. This can be prepared ahead.
Preheat the oven to 250 degrees F.
In a large, deep-sided pan over high heat, add 1/4 cup of the garlic oil. When it just starts to smoke, add the sliced garlic cloves and cook until just crisp and light golden brown, 30 seconds. Remove to a paper towel-lined plate and sprinkle with a touch of salt. Add in the chicken and carefully brown the pieces, turning as needed, and getting a nice even dark golden brown on all sides. As the pieces finish, remove to a baking sheet and hold warm in the oven.
Drain the excess fat from the pan the chicken was cooked in and set heat at medium-high. Add in the remaining 1/4 cup garlic oil and the flour and stir constantly until a paste forms. Allow the roux to cook for 1 to 2 minutes, with continued stirring, and then slowly whisk in the stock. Add about 1/2 cup at a time and continue to incorporate. Reduce the heat to low once the mixture has thickened and then add in the reserved cooked garlic cloves.
Whisk these into the roux, adjust seasoning with salt and pepper, if needed, and hold warm until service over low heat.
Place 1/2 cup of the garlic gravy on a plate and position a couple pieces of chicken on top. Spoon 1 tablespoon of the gravy on top of the chicken and garnish with the parsley, diced tomatoes, and the fried garlic slices.
In a small saucepan over medium heat, add half of the oil and all of the garlic. Heat through and cook for 10 minutes, adjusting the heat as needed. Allow to cool to room temperature.
In a 16-ounce clean glass jar, add the prepared garlic and the remaining oil. Cover and refrigerate. Use within 2 to 3 weeks.
Use a good quality olive oil in this, but not extra-virgin. You want an olive oil that can withstand some heat, as you will soon be cooking everything with this oil!
Find the freshest, firmest, whitest garlic bulbs you can find. Gently break apart the bulb, and then carefully trim the stem end from each clove and peel the skins. Be picky, use the biggest ones and leave the smaller ones for another recipe.
Use the oil for cooking, use the garlic for everything. Once the garlic is depleted, start over.