Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 10x15 inch jellyroll pan. Set aside 1/2 cup of the dry cake mix.
In a medium bowl, mix together eggs and sour cream. Stir in the package of cake mix. Batter will be lumpy. Spread batter into the prepared pan. Using the back of a spoon, make 15 wells in the batter, 3 rows of 5. In another bowl, beat together the cream cheese and milk until fluffy. Place one tablespoon of the cream cheese mixture into each well. Place 1 apricot half, cut side up, onto each blob of cream cheese.
In a small bowl, combine the 1/2 cup of reserved cake mix with the butter. stir in the butter until the mixture is crumbly. Stir in the slivered almonds, and sprinkle the mixture evenly over the Danish.
Bake for 30 to 35 minutes in the preheated oven, or until a toothpick inserted into the cake comes out clean. To make the glaze, stir together the confectioners' sugar and water until smooth, adding another teaspoon of water if necessary. Drizzle over the Danish when cool.
