For the pudding:

1. In a large bowl, combine the cream and milk.
2. In a large heavy saucepan, combine the brown sugar, water, and salt, and place over medium-high heat. Bring to a boil over medium-high heat and continue to cook for 10-12 minutes, stirring occasionally to make sure the mixture doesn’t scorch. When the mixture is deep brown and smells nutty, remove from the heat and immediately whisk in the cream mixture. The mixture will steam vigorously, and the sugar will seize. Use caution to avoid being burned by the bubbling mixture. Whisk until smooth and the caramel is fully incorporated. Return to high heat, bring to a boil, then turn off the heat.
3. In a bowl, whisk together the egg, yolks, and cornstarch. While whisking constantly, add about half of the caramel cream , ½ cup at a time, to the egg mixture. Pour the combined mixtures back into the saucepan and add the remaining caramel cream. Cook over medium heat, whisking constantly, for about 2 minutes, or until the custard is very thick.
4. Remove the custard from the heat and whisk in the butter and rum. Pour the custard through a fine-mesh sieve into 8 to 10 three-fourths-cup ramekins or small glasses, dividing it evenly. Cover with plastic wrap, and refrigerate until well chilled, at least 2 hours, or up to overnight.

For the rosemary caramel sauce:

1. Combine the brown sugar, butter, and half and half in a small saucepan. Bring to a boil over medium heat, cooking until thickened, 1-2 minutes.
2. Remove from the heat and add the rosemary, muddling sprigs slightly.
3. Steep for 10-15 minutes, and then remove rosemary. Keep warm.

To serve:

1. Just before serving, whisk the cream in a chilled medium bowl until it begins to thicken. Add the crème fraiche and continue whisking until thick and fluffy.
2. Top each portion of the budino with some of the whipped cream mixture and a few peach slices, then spoon a bit of warm rosemary caramel over each portion and serve.