Prepare an 8 x 12-inch pan by greasing and generously dusting with confectioners’ sugar. For mini marshmallows, use a half sheet pan, 18 x 13-inch. For big, square marshmallows, use an 8 x 8-inch pan.
Sprinkle unflavored gelatin over 3/4 cup of cold water in the bowl of an electric mixer. Set aside while making the syrup.
In a medium saucepan, combine granulated sugar, corn syrup, salt and 1/2 cup water. Cook over medium heat until mixture reaches 240°F, soft ball stage.
Using the whisk attachment and with the mixer on low speed, slowly pour the hot sugar syrup into the dissolved gelatin. Turn the mixer to high speed and whip until very thick. This will take about 15 minutes. Add vanilla and peppermint extract and mix thoroughly.
Pour mixture into the prepared pan, using a lightly oiled spatula. Spread evenly, and lightly press half of the crushed peppermint candies into the top of the mixture. Dust top with powdered sugar and let stand uncovered overnight allowing it to dry out.
Turn marshmallows onto a cutting board. Using a lightly oiled knife or scissors, cut into squares. Press peppermint pieces into the sides of the marshmallows. Dust with more powdered sugar to keep them from sticking together.
Store any leftovers in an airtight container for up to one week.