Preheat oven to 350 degrees. In a medium bowl, combine oats, flour, baking powder, and salt; set aside.
In the bowl of an electric mixer, cream butter and sugars on medium speed until smooth; beat in egg and vanilla. With mixer on low speed, add oat mixture, and mix until just combined. Stir in apricots and almonds by hand.
Divide dough into twelve pieces, and roll into balls. Place balls 3 to 4 inches apart on baking sheets. Using your fingers, flatten balls to about a 3/4-inch thickness.
Bake cookies until edges begin to brown, 25 to 30 minutes. Let cool 5 minutes on baking sheet, then transfer to a wire rack to cool completely.