Line a baking sheet with waxed paper. Put pistachios in a shallow bowl.
Melt chocolate (see "Melting Chocolate: Be Gentle," below). Remove from heat. Holding apricots from one end, dip them 1 at a time into the chocolate, turning to coat about 3/4 of the apricot. Use a knife to scrape excess chocolate off the bottom back into bowl. Hold apricot over bowl of pistachios and sprinkle the nuts over the chocolate.
Put apricot (nut side down) on baking sheet and repeat with remaining apricots. Chill until chocolate is set, about 1 hour. Store in an airtight container between sheets of waxed paper in the refrigerator for up to 2 weeks.
Melting chocolate: Be gentle. Fill a saucepan with 2 to 3 in. of water. Put finely chopped chocolate in a heatproof bowl that will fit over pan (the bottom of the bowl should not touch the water); set bowl aside. Bring water to a boil, turn off heat, then set bowl over pan. Use a heatproof spatula or wooden spoon to stir the chocolate frequently and encourage even melting. Once the chocolate is melted, stir well and use immediately.
Note: Nutritional analysis is per apricot.