Step 1Wash the game hens or chicken and dry thoroughly. Remove the backbone from each bird (cook with the other pieces, or save for another use), and cut the remaining chicken into 8 pieces (2 breasts, 2 wings, 2 thighs and 2 legs -- 9 if you include the backbone). If the breasts are large, cut each in half crosswise to make a total of 10 pieces.
Step 2In a deep, medium bowl, combine the kosher salt, lemon zest and juice, minced thyme leaves, several grinds of black pepper and the garlic to form a rub. Add the chicken pieces to the bowl, massaging the rub all over each of the pieces. Cover the bowl with plastic wrap and refrigerate overnight, or at least several hours.
Step 3The next morning, pour the buttermilk over the pieces and gently toss to coat; the buttermilk should barely cover the chicken; if not, add just enough to roughly cover. Cover the bowl again and refrigerate for 4 to 6 hours for game hen, 6 to 8 for chicken.
Step 4Season the flour: Place the flour in a large bag, bowl or baking dish and season with 1 1/2 teaspoons table salt and 1 1/2 teaspoons freshly ground pepper. Taste the flour, and adjust seasoning if desired.
Step 5About 1 hour before frying, remove the bowl from the refrigerator. Remove each piece of chicken from the buttermilk, shaking gently to remove any excess buttermilk (do not attempt to dry the pieces). Dredge each piece in the seasoned flour mixture, coating completely. Shake to remove the excess flour, and set the pieces aside on a rack to dry and warm to room temperature.
Step 6While the pieces are resting, prepare the lard: Place about 4 cups lard in a large heavy skillet or frying pan over medium heat. Melt the lard; it should come up about half to three-fourths of an inch up the side of the pan (melt additional lard if needed). When the lard is just melted, add the onion rings and continue heating the lard until the onion is caramelized and the lard is hot. Remove the onion (discard it, or save for another use), and check the temperature of the lard; a thermometer inserted should read 350 degrees.
Step 7Gently place the pieces, skin-side down, in the hot lard, being careful not to crowd. Lower the temperature to 325 degrees and fry the pieces on each side until crisp and golden brown and the meat is firm and opaque, about 5 minutes for game hen pieces and 7 to 8 pieces for chicken. Flip the pieces over and fry on the other side until done (a thermometer inserted in the meat should read 160 degrees). Remove the pieces from the hot oil and drain, skin-side up, on crumpled paper towels. Repeat until all of the pieces are fried.
Step 8Serve the pieces hot or at room temperature.