Preheat the oven to 325°F. Line a baking tray with parchment paper.
In a large bowl whisk the egg white until frothy. Add the pecans and stir to coat. Add the brown sugar, ¼ teaspoon of the cinnamon, ¼ teaspoon salt, and the cloves. Stir until evenly distributed among the nuts. Spread the nuts in a single layer on the prepared pan and bake until toasted and caramelized, 18 to 20 minutes, stirring several times. Once cool enough to handle, coarsely chop the pecans and set aside.
Prepare the grill for indirect cooking over medium heat (350° to 450°F).
In a small bowl combine the butter, the remaining ½ teaspoon cinnamon, ¼ teaspoon salt, and the cayenne pepper. Place the squash, cut side up, on the grill and brush generously with the butter mixture, reserving the rest for basting and serving. Grill over indirect medium heat, with the lid closed, until very tender, 50 minutes to 1 hour, basting occasionally. Remove from the grill and drizzle with the remaining butter mixture. Top with candied pecans and a final sprinkling of salt, if desired. Serve warm.