1. Combine 1/2 cup ice and basil in a cocktail shaker; muddle. Add remaining 1/2 cup ice, vodka, and vermouth to shaker. Cover and shake 30 seconds. Strain vodka mixture evenly into 4 martini glasses.
2. Make a tunnel in each tomato using a cherry pitter or skewer. Stuff each with 1/2 teaspoon blue cheese. Place 1 stuffed tomato in each glass.