Use neutral vodka for more basil flavor.
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.
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.
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