Prepare the grill for indirect cooking over low heat (300° to 325°F). Set up the rotisserie according to the manufacturer’s instructions.
Combine 1 tablespoon of the salt, 2 teaspoons of the pepper, the granulated garlic, and sage. Trim the thick fat cap on the roast, leaving very little, if any, exterior fat. Using butcher’s twine, tie the roast firmly in three places into a cylindrical shape (this will help it cook evenly). Brush the roast all over with the oil and season with the spices.
Center the roast on the rotisserie spit and secure it with the spit forks. Make sure the roast is centered before tightening the forks completely. Place the spit into the motor and turn it on. Place a disposable foil pan under the roast on the cooking grates to collect the drippings. Grill the roast over indirect low heat, with the lid closed, until an instant-read thermometer inserted into the center of the meat, not touching the metal spit, registers 125° to 130°F for medium rare, 1½ to 1¾ hours.
About 25 minutes before the roast is ready to serve, prepare the mushrooms and onions. In a large nonstick skillet over medium-high heat on the stove, melt 4 tablespoons of the butter. Put the mushrooms (rounded side down first) and onions in the skillet and season with the remaining ½ teaspoon salt and ½ teaspoon pepper. Cook until the mushrooms and onions are golden brown, 8 to 10 minutes, turning once after 4 to 5 minutes. Reduce the heat to medium-low, cover the skillet, and cook until the onions are tender and cooked through, 12 to 15 minutes more, turning occasionally for even cooking. Add the cognac and lemon juice to the skillet; increase the heat to medium-high and cook, uncovered, until the liquid is reduced by half, 30 seconds to 1 minute. Add the broth and cook until the liquid is reduced by about one-third, 2 to 3 minutes. Stir in the remaining 4 tablespoons butter and swirl the skillet, forming a slightly thickened sauce. Remove from the heat and cover to keep warm.
When the roast is done, transfer to a cutting board and let rest for about 20 minutes (the internal temperature will rise a few degrees during this time). Remove the roast from the rotisserie spit and cut away the twine. Carve the roast into thick slices and serve warm with the mushrooms and onions.