Preheat the oven to 350ºF. In a medium bowl, stir together the flour, baking powder, and salt until evenly mixed.
In a large bowl, beat together the sugar, butter, eggs, vanilla extract, almond extract, and nutmeg until the mixture is light and creamy in appearance.
Pour the dry ingredients into the bowl of wet ingredients and stir until a soft dough forms.
Chop the apricots into small pieces. Add the chopped apricots and sliced almonds to the bowl with the dough, then stir until they are mixed in. Save a few pieces of almond to press into the top of the biscotti.
Line a baking sheet with parchment paper. Divide the dough into two equal pieces, then shape each into a flattened log, about 3 inches wide and 12 inches long. Place the logs a few inches apart on the baking sheet. Press a few pieces of almonds into the top of the dough for decoration. Bake the biscotti logs for 35 minutes in the preheated oven, or just until the edges turn golden and the top has a few cracks.
Remove the biscotti from the oven, but don't turn it off. Let the biscotti cool for about five minutes, then carefully transfer them to a cutting board. Use a serrated knife to cut the biscotti, on an angle, into 3/4-inch wide slices. Place the slices on back on the baking sheet, cut sides up.
Bake the sliced biscotti for about five more minutes, then flip and bake for five minutes on the second side. Allow the biscotti to cool completely before serving or packing in an air-tight container.