Process 1/2 cup raspberries in a blender or food processor until smooth, stopping to scrape down sides. Pour through a wire-mesh strainer into a bowl, discarding solids. Set aside.
Beat egg whites at high speed with an electric mixer until stiff peaks form. Gradually add 1/4 cup sugar, beating at low speed until blended; set aside.
Cook remaining 1 cup sugar and 1/4 cup water in a heavy saucepan over low heat, without stirring, until sugar dissolves and a candy thermometer registers 248° (about 10 minutes). Remove from heat.
Pour hot sugar syrup in a thin stream over egg whites, beating constantly at high speed for 5 minutes. Gradually add butter pieces and raspberry puree, and beat frosting mixture at medium speed 2 minutes or until creamy.
Spoon frosting mixture into a 1-quart zip-top plastic freezer bag or piping bag. Seal freezer bag, and snip a tiny hole in 1 corner of bag. Pipe frosting on cut sides of ladyfingers, and top evenly with sliced strawberries. Dust with powdered sugar.
Note: Frosting mixture may be stored in an airtight container in the refrigerator for up to 1 week. To soften for piping or spreading consistency, place in a glass bowl; microwave at HIGH 45 seconds. Whisk and then microwave 30 more seconds. Whisk until smooth.