Line a baking sheet with parchment paper; set aside. Prepare an ice-water bath. Insert one 6-inch-long 1/4-inch dowel into the top of each apple. Place cream, sugar, butter, and corn syrup in a heavy-bottomed saucepan, and bring to a boil over medium heat. Continue cooking until the temperature registers 245 degrees on a candy thermometer, 10 to 12 minutes. Remove from heat, and briefly plunge the saucepan into ice water to stop the caramel from cooking.
Dip one apple into the caramel, and coat the top and sides with caramel using a spoon. Transfer to prepared baking sheet to cool. Repeat with the remaining apples.
Dip a whisk or fork into the sugar syrup, and wave back and forth over an apple to draw out long, fine, threadlike strands. Gather the ends of the strands to form a weblike shape around each apple. Repeat with remaining apples, and serve.
Instead of a fork, Martha uses a cut-off balloon whisk to spin the sugar webs. When working with hot sugar, it is always a good idea to keep a bowl of ice water nearby in case you burn yourself. Humidity causes spun sugar to soften and disintegrate, so be sure to work in a cool, dry environment.