Preheat the oven to 350°F. Grease an 8x8 inch baking dish. Sprinkle a teaspoon of granulated sugar on the bottom of the baking dish, and set aside.

In a small bowl, combine the light brown sugar and ground cinnamon. Mix together, and set aside.

Open the can of crescent rolls and gently spread the dough out on a flat surface. With a sharp knife, score the perforated lines. Take the brown sugar-cinnamon mixture, and sprinkle over the dough, evenly spreading it out.

Starting from the widest edge of the dough, roll up a peach slice and gently secure the edges. Place flat in the baking dish. Repeat this process with all peach slices and crescent rolls.

Next, in a small saucepan, melt the butter over medium heat. Once melted, add the half cup granulated sugar and the vanilla extract. Whisk together until well-combined. Evenly pour the mixture over the top of the dumplings. Finally, pour the lemon-lime soda and pour it around the edges of the pan.

Bake for approximately 40 minutes, until the tops of the dumplings are golden brown.

While the dumplings are baking, make the brown butter caramel sauce. In a small saucepan over medium-low heat, melt the unsalted butter and allow it to foam. 

Swirl the pan from time to time to keep the butter from burning. Once it begins to turn amber and has a nutty fragrance, remove it from the heat.

Pour the browned butter into a bowl and set it aside to cool.

Thoroughly wash the saucepan, and dry it, before continuing to make the sauce. Heat the clean, dry pan over medium heat and add the sugar. Swirl the pot occasionally as the sugar melts. Continue to do this until the sugar is completely melted, about 5 minutes.

Next, whisk in the browned butter and remove the pan from the heat. Carefully drizzle the heavy cream into the sugar mixture, whisking simultaneously. CAUTION: The mixture will bubble violently, but continue to whisk through this reaction. The bubbling will subside, and the caramel will come together. Add the pinch of sea salt, and whisk again. Remove the caramel from the heat completely, and allow it to cool while the dumplings finish baking.

Once the dumplings are done, serve immediately, with ice cream if desired, and a hearty drizzle of the caramel sauce.