Preheat oven to 325F. Butter a 12-well standard muffin tin.
In a medium sauce pan over medium heat, melt the butter. Add the sugar and stir until the sugar is melted and the butter and sugar are well combined into a sauce. Divide the sauce among the 12 wells of the muffin tin.
Fill a bowl with ice water. Bring a large pot of water to a boil. Drop in the apricots and let them boil for 1 minute. Immediately transfer them to the ice water. Once they are cool, peel off the skins. Carefully cut each apricot in half (THEY ARE SLIPPERY!) and remove the pit, and then slice each half into two round slices. Each apricot will yield 4 slices. Put one slice into the bottom of each muffin well.
In a medium bowl, mix together the flour, baking powder, salt, and cinnamon.
In a large bowl, whisk the eggs vigorously until they are light colored and frothy and full of tiny bubbles. Whisk in the jam and zest, then whisk in the orange liqueur. Add the dry ingredients and whisk in.
Divide the batter among the muffin wells. Tap the tin sharply a few times to help the batter settle to bottom and remove air pockets around the apricot slice. Bake the cakes for 18-22 minutes.
Transfer the tin to a cooling rack and let the cakes rest for about 10 minutes. Run a knife or spatula around the edge of cake to loosen it from the muffin tin. Then carefully invert the tin onto the cooling rack by putting the cooling rack over the top of the tin and then flipping them both over. Serve cakes as is, or with any combination of orange liqueur, powdered sugar, and vanilla ice cream.