Pluck the basil leaves from the stem and cut into small pieces. Mix the sour cream, crème fraîche, cream, lemon juice and basil together and place in an ice cream maker. If there is no ice machine available, put everything in a bowl in the freezer, check several times and stir with a whisk (later with a fork) to destroy the ice crystals.
Caramelize the muesli together with the sugar in a pan and arrange on four plates as a bed for the peach.
Halve and core the peaches. Sprinkle the cut surfaces with a little brown sugar and grill first on the round, then on the cut surface. Then arrange on the muesli bed and place a scoop of sour cream next to it.