Combine granulated sugar and 1/3 cup water in a medium, heavy saucepan over medium-high heat; cook until sugar dissolves, stirring gently as needed to dissolve sugar evenly. Continue cooking 8 minutes or until golden brown (do not stir); gently tilt pan and swirl mixture to evenly brown. Remove from heat. Slowly add cider, amaretto, and butter, stirring constantly with a long-handled wooden spoon (mixture will bubble vigorously). Cook over medium-low heat 5 minutes or until smooth. Stir in pecans. Remove from heat.
Sprinkle cut sides of peaches evenly with brown sugar and salt. Place peaches, cut sides up, on a grill rack coated with cooking spray; grill 2 minutes on each side or until tender. Serve warm with sauce.