Preheat the oven to 350°F (180°C). Prepare a rimmed baking sheet lined with a sheet of parchment paper.

To toss peaches in a large bowl, mix lime juice made out of two limes, granulated sugar, 7 long strips made by peeling the lime skin, and rosemary sprigs. Add peaches, stoned, and cut into 1/2 cm-thick slices (about 10.5 oz. or 300 g net weight).

Soak peaches in the lime and rosemary-infused syrup at room temperature for 40 minutes to 2 hours. Drain peaches with a colander over a small bowl, and discard the rosemary and lime peel. You should get about 2 fl. oz. (60 ml) of peach syrup. Set it aside and keep it for later.

Roll out the puff pastry to a thickness of 1/5 inch/1/2 cm into a ten-inch/26 cm-wide circle using a rolling pin. Then place on a prepared baking sheet with parchment. Arrange strained peach wedges in a single layer on the bottom of the crust. Place them haphazardly or go around the circle like for an apple pie. Leave a clear 2-inch border all around the edge.

Then fold these 5 cm edges of the dough over the peach slices, slightly covering the fruit. Place small butter cubes over fresh fruit. Brush the folds of the puff pastry all over with the beaten egg. Sprinkle sugar over the folds of the pastry and bake for 20 minutes. Watch out for the golden edge of the crust and the bubbling of the peach filling.

While the galette is baking, whisk cornflour into the reserved peach syrup. Pour it into a small saucepan and heat over medium-high heat for 2 minutes until it thickens (the consistency should resemble honey).

Then pour this preparation over half-baked peaches. Sprinkle the rosemary leaves on top of the peaches. Bake the galette for the other 15 minutes until the pastry is golden brown and the filling is bubbling. 

To make the lime whipped cream, whisk whipping cream with sugar and lime zest, and refrigerate till serving. Take the galette out of the oven, and let it cool slightly. Serve it with lime-infused whipped cream on top.