Combine the apples, pears, lemon juice, cinnamon sticks, nutmeg, 1 tablespoon sugar and 1/4 cup water in a medium saucepan. Bring to a boil over medium-high heat. Cover, reduce the heat to medium-low and cook until the apples and pears are very soft but still a bit chunky, 15 to 20 minutes. Taste the mixture and add the remaining tablespoon of sugar if desired.
Remove the sauce from the heat and stir a few times until the sauce is well combined with small chunks. Cool to room temperature. Remove the cinnamon sticks and serve. (For a smoother sauce, remove from heat, remove the cinnamon sticks and then whisk the mixture until smooth before cooling.)