Liberally grease a 9 x 9 pan and set aside
Place 3/4 cup apple cider in a saucepan and turn heat to medium. Bring to a boil and then lower to a very high simmer until the cider reduces to 1/4 cup. This should take around 10 minutes
Place sliced apples in a glass bowl, cover with filtered water and squeeze the juice of one lemon into the water. Let apples soak for 5 minutes. Remove from water and lay slices on a paper towel. Pat dry and set aside
Place butter and sugar in a mixer and mix until creamy
Add eggs and beat until just mixed
Add flour, baking powder, baking soda, salt, reduced apple cider, and half & half. Beat until just mixed
Preheat oven to 350 degrees F
Use a spatula to transfer the cake batter to the 9 x 9 pan and stick each apple slice into the batter. In a small bowl mix the cinnamon and sugar and sprinkle it over the apples
Bake for 40 minutes. Test at that point and if they aren't done, place in there for 5 more minutes. Place on a wire rack when done and let cool for 15 minutes before adding the glaze
As the cake is baking, reduce another 3/4 cup of apple cider to 2 tablespoons. (Follow the directions above)