Prepare the Grill
Begin by setting up your charcoal grill for two-zone cooking, with the coals concentrated on one side for direct heat. If using a gas grill, heat all burners to medium-high, then switch off one of the end burners to create indirect heat.
Prepare the Brine and Fennel
While the grill is heating, pat the salmon fillets dry and set them aside to air-dry. In a metal 3-quart dish, such as a 9x13-inch pan, or a large oven-safe skillet, mix together the lime juice, vinegar, olive oil, and sugar. Add the sliced fennel, thyme (or oregano), and garlic. Season generously with salt and pepper, then stir to combine all ingredients.
Warm the Brine
Once the grill has reached temperature, place the dish of fennel mixture over the indirect heat zone of the grill. Cover and allow it to warm for about 5 minutes.
Grill the Salmon
Season the salmon fillets with salt and lightly coat with olive oil. Prepare a paper towel soaked in olive oil and a fish spatula, and bring a rimmed baking sheet to the grill. Using gloved hands, transfer the dish of fennel brine to the baking sheet or another heat-safe surface.
Grill the Salmon Fillets
Clean and oil the grill grates using the prepared paper towel. Place the salmon fillets skin-side down over the direct heat. Close the grill lid and cook until the skin is browned and releases easily from the grates, and the flesh turns opaque about three-quarters of the way up the sides—approximately 3 to 6 minutes, depending on thickness. Be sure to check the salmon frequently and move it around the grill as needed to prevent flare-ups.
Finish the Dish
Once the salmon is nearly cooked through, carefully place it skin-side up into the warmed fennel brine. Allow the dish to rest for 5 minutes, or up to 30 minutes, until the salmon is fully cooked. Serve the salmon alongside the pickled fennel. Leftovers can be stored in the refrigerator and enjoyed cold or at room temperature the following day, though the skin should be discarded as it will become soggy after refrigeration.