Peel, halve and core the apples. Then cut into small bite-size chunks.
Juice the lemon and pour the juice into a large bowl.
Add the apples to the bowl with the lemon juice and toss together.
Add the sugar to the apple mixture, a 1/3 cup or so at at time, and mix the apple mixture to evenly coat until the full cup of sugar has been mixed into the apples.
In a large skillet, melt the butter over a medium-high heat.
Add the apple mixture to the butter - stirring occasionally. Bring to a simmer, checking that the juices all run clear. This is means the sugar has dissolved and incorporated into the liquid. This will take about 2 minutes.
Cover and turn the heat down to medium-low. Cook for about 7-10 minutes longer to soften the apples. Stir once or twice to ensure the apples all cook evenly. You want to leave a little bit of a crunch in them, but they should be soft.
Place a strainer over a medium sized bowl and pour the apple mixture into the strainer to separate the liquid from the apples. Toss the strainer a bit to ensure all of the liquid is separated and place the apples in a large bowl.
Pour the liquid back into the skillet and simmer over a medium heat for 10-15 minutes. The liquid should thicken and reduce down a bit, and will lightly caramelize.
Sprinkle the cinnamon and nutmeg over the top of the apple mixture.
Pour the caramelized liquid over the apple mixture. Toss the apple mixture to evenly mix the spices and liquid. Set aside (or refrigerate) to cool.