Heat oven to 350°F. Line two baking sheets with parchment.
Beat butter and sugar until pale and fluffy, about 2 minutes.
Mix in vanilla and salt until combined.
Add flour and cornstarch. Mix just until the dough looks sandy and clumps together.
Press dough into a smooth mass by hand.
Roll between parchment to 1/4 inch thick. Chill 10 minutes for cleaner cuts.
Cut 2-inch rounds or scoop 1 tablespoon mounds. Arrange 2 inches apart and flatten lightly.
Bake 11 to 13 minutes, until set and barely golden at the edges.
Cool on pans 5 minutes. Transfer cookies to a rack to cool completely.
Microwave caramels with cream in 30-second bursts, stirring, until smooth and pourable.
Spoon about 1 teaspoon caramel onto each cookie center. Let stand until tacky, 10 to 15 minutes.
Melt chocolate with oil until glossy. Spread over caramel. Sprinkle flaky salt. Let set completely.