Preheat oven to 375 degrees F (190 degrees C). Grease a 10-cup fluted tube pan (such as Bundt(R)).
Combine 1/2 cup sugar, egg, and butter in a stand mixer fitted with the paddle attachment; beat until smooth. Beat in vanilla extract.
Mix flour, baking powder, and salt together in a separate bowl.
Alternate adding flour mixture and milk to the creamed butter mixture, beating well after each addition. Fold apricots gently into the batter.
Mix 3 tablespoons sugar and cinnamon together in a small bowl. Sprinkle 1/3 of the cinnamon sugar into the greased pan. Fill pan halfway with batter; sprinkle 1/2 of the remaining cinnamon sugar on top. Pour in remaining batter; cover with remaining cinnamon sugar.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 45 minutes.