Combine 1/2 cup granulated sugar and 2 tablespoons water in a medium saucepan. Cook over medium heat, stirring occasionally until the sugar dissolves. Once dissolved, stop stirring and let the mixture boil until it turns a deep amber color, swirling the pan gently if needed (about 5-8 minutes).

Carefully and slowly pour the warmed heavy cream into the caramel while whisking continuously (it will bubble vigorously). Remove from heat and whisk in the cubed butter and 1/2 teaspoon flaky sea salt until smooth. Pour into a small heatproof bowl and let cool completely at room temperature. This can take 30-60 minutes.

In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.

In the bowl of a stand mixer fitted with the paddle attachment (or using a hand mixer), cream together the softened unsalted butter, 1/2 cup granulated sugar, and light brown sugar on medium speed until light and fluffy (about 2-3 minutes).

Beat in the large egg until just combined, then mix in the vanilla extract.

Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined and no streaks of flour remain. Do not overmix.

Line a baking sheet with parchment paper. Using a 1.5-inch cookie scoop, scoop out portions of cookie dough. Divide each scoop in half. Roll one half into a ball, then flatten it into a disc.

Spoon about 1/2 teaspoon of the cooled salted caramel onto the center of the flattened disc. Take the other half of the cookie dough and flatten it, carefully placing it over the caramel-filled disc. Pinch and seal the edges completely, then gently roll into a smooth ball, ensuring no caramel leaks out. Place on the prepared baking sheet.

Repeat with the remaining dough and caramel. Chill the assembled cookie dough balls in the refrigerator for at least 30 minutes. This helps prevent spreading during baking.

Preheat your oven to 375°F (190°C). Line two baking sheets with parchment paper.

Place the chilled cookie dough balls about 2 inches apart on the prepared baking sheets (around 12 cookies per sheet).

Bake for 12-14 minutes, or until the edges are golden brown and the centers are just set. The cookies may still look slightly underdone in the middle, but they will continue to set as they cool.

Remove from the oven and immediately sprinkle with additional flaky sea salt, if desired. Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.