Bring a large pot over water to a boil over high heat. Prepare an ice-water bath; set aside. Cut a small X into the bottom of each tomato. Add tomatoes to boiling water for 10 seconds. Immediately transfer tomatoes to ice-water bath using a slotted spoon; let cool. Peel and core tomatoes.
Place basil sprigs and garlic slices in the bottom of a small baking dish. Top with tomatoes, core side down. Sprinkle with salt; drizzle with olive oil.
Transfer tomatoes to oven and bake until lightly browned and tender, about 50 minutes. Let tomatoes cool before transferring to an airtight container. Cover tomatoes with olive oil and refrigerate, covered, up to 5 days.