Add butter and vanilla; cut in using on/off turns until butter forms pea–size pieces.
Add milk and blend in using on/off turns until mixture forms small moist clumps. Gather dough into ball; flatten into disk. Wrap dough in plastic; chill 1 hour.
Roll dough out between sheets of floured parchment to 12–inch round. Press dough onto bottom and up sides of 10–inch–diameter tart pan with removable bottom. Fold dough edges over and press to form thick sides. Using thumb, press dough around sides to extend crust 1/8 inch above edge of pan. Freeze 20 minutes.
Preheat oven to 350°F.
Bake crust until brown, piercing bubbles with fork, about 30 minutes.
Transfer to rack; cool.
Bring cream and corn syrup to simmer in medium saucepan; remove from heat.
Add all chocolate; whisk until smooth. Cool 30 minutes.
Pour filling into crust.
Sprinkle with 1/2 cup candy. Chill until set, about 3 hours. Do ahead
Can be made 1 day ahead. Cover and keep refrigerated.
Using electric mixer, beat all ingredients in large bowl to stiff peaks.
Spread 2 cups whipped cream over filling, mounding slightly in center. Spoon remaining whipped cream into 1 bottom corner of large resealable plastic bag. Twist at top; grasp top firmly. Using scissors, cut 3/4 inch off filled corner to form opening; turn bag so that 1 seam faces up. Pipe 2–inch–long ovals of whipped cream side by side around edge of tart. Pipe more ovals to form second ring inside first, then pipe ovals in center. Chill at least 30 minutes and up to 3 hours.
Remove tart from pan; place on platter.
Sprinkle top of tart with remaining 1 tablespoon crushed peppermint candies and serve.
