Cut off the stem of the eggplant and peel. Slice the eggplant into 1/2-inch thick disks and arrange them in a single layer on a baking sheet. Sprinkle with salt and let sit for 20 minutes. This step draws the bitterness out of the eggplant.

Use a clean towel to blot the eggplant dry. Cut the disks into 1/2-inch sticks and place them in a medium bowl.

Meanwhile, heat the tallow in a large skillet over medium-high heat. Sprinkle the arrowroot over the eggplant and toss until all pieces are evenly coated. Carefully place the eggplant into the hot oil and fry until golden brown on the bottoms.

Using tongs, turn each piece over and continue frying until the other side is golden brown. Do not overcrowd the pan—fry in a single layer for best results. Transfer the fries to a baking sheet lined with a cooling rack (this keeps all sides crispy). Drizzle with honey, lightly season with salt, and sprinkle with fresh thyme. Serve immediately.