Make the pickled shrimp and tomatoes:
Bring a 4-quart saucepan of water to a boil over high heat. Have ready a large bowl of ice water.
Score the bottom of each tomato with a small X. Blanch the tomatoes in the boiling water until you see the skin near the scoring lift, 30 to 60 seconds. Using a slotted spoon, transfer the tomatoes to the ice water to cool. Remove the tomatoes from the ice water and peel. Reserve the ice water, adding more ice if the first batch has melted.
In a 2- to 3-quart saucepan, bring 1-1/2 cups water and the vinegar, sugar, thyme, horseradish, coriander, pepper flakes, lemon, and 1/2 tsp. salt to a boil over medium heat. Reduce the heat to a bare simmer. Add the shrimp first, then the peeled tomatoes. Cook, stirring constantly, until the shrimp are pink and just cooked through, about 2 minutes. Plunge the bottom of the saucepan into the ice water and let sit, stirring constantly, until the shrimp are cool enough to touch, about 5 minutes. Transfer the mixture to a container and refrigerate, uncovered, until cold, about 2 hours. Cover.
Make the cocktails:
Strain the shrimp and tomato mixture through a medium-mesh sieve set over a 1-quart measuring cup.
For each cocktail, fill a mixing glass three-quarters full with ice. Add 2 fl. oz. (1/4 cup) gin, 1 fl. oz. (2 Tbs.) vermouth, and 4 fl. oz. (1/2 cup) of the shrimp pickling liquid. With a long-handled spoon, stir until well chilled, 25 to 30 rotations. Strain into a chilled martini glass. Skewer 1 pickled shrimp and 1 tomato on a toothpick and add it to the drink. Repeat to make 5 more drinks.
The pickled shrimp and tomatoes can be prepared up to 5 days ahead. Refrigerate them in their pickling liquid, covered.