Peel, core, and slice the cooking apples, then place in a wide pan with the sugar and butter. Cook over high heat for 10 to 15 minutes until the apples have broken down to a pulp and an excess water has cooked off. Transfer to a bowl and cool completely (you should have about 1 lb of puree).
Heat the oven to 325°F Line and lightly grease a 9-inch cake pan with a removable bottom. Peel, core, and finely slice the Braeburn apples, using a mandoline or sharp knife. Place in a bowl and pour over most of the lemon juice and a splash of water, then set aside.
In a large bowl, mix together the flour, baking powder, baking soda, salt, sugar, and ground spices. Make a well in the center and add the egg, olive oil, and apple puree. Fold into dry ingredients until just combined.
Transfer the mixture to the prepared cake pan and gently level the top with a spatula. Bake the cake for 30 minutes until it feels just firm to the touch in the center.
Working quickly, overlap the sliced apples in concentric circles on rop, leaving a margin around the edge. Brush the slices with a little lemon juice and return to the oven for anoher 30 - 35 minutes until a skewer inserted in the middle comes out clean.
Let the cake cool slightly before unmolding onto a rack. Warm the apricot jelly with the water, stirring until smooth. Brush over the top of the cake to glaze and serve warm.