1)  peaches and brut champagne will give me this cocktail.....

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actually the correct answer would be a "peach bellini" as many different fruits can be used in an bellini and if you wan't to go a step further you can use any fruit sorbet then the cocktail would be called a sorbellini 
                                                                                       

Peaches in Brut Champagne


Ingredients :

1/2 bottle brut champagne
2 ripe peaches
1 dl whiterum 4-5 tsp sugar

Recipe :
1 - Put the Peaches in a small saucepan, cover and bring to the boil. Simmer gently for 5 mins.
2 - Remove the Peaches and peel. Pour away the water.
3 - Put the sugar in the saucepan and pour over half and glass of water. Heat gently for 10 mins, until a syrup is formed.
4 - Add the peachesand stir. Finally pour in the rum.
5 - Remove from the heat and allow thepeaches to cool.
6 - To serve, place the peaches in 2 large glasses, with the syrup. Cover withchampagne.


Advice :
Prepare the peaches a little in advance, so they have time to cool down.

Bellini

Ingredients :

Large measure of fresh peach juice 

Champagne or good sparkling wine. Italians traditionally use prosecco, champagne makes a better blend.

Chill both ingredients thoroughly and pour a large measure of peach juice into a tall tulip glass. Top up with champagne.




Bellini drink you use Large measure of fresh peach juice and not the actual peach in drink or are you throwing a curve again puzzykat?

Bellini you also can use any brand of champagne!!!

To take your description literally because you use the whole peach it is called Peaches in Brut Champagne !!!!   or just  Peaches in Champagne