In a large saute pan, over medium heat, melt the butter. Stir in 3/4 cup of the sugar until it dissolves. Add the peaches and saute for 3 to 4 minutes, or until the peaches soften. Remove from the heat and cool completely. In a food processor, fitted with a metal blade, puree the peach mixture until smooth. In a medium saucepan, over medium heat, combine the peach mixture and heavy cream. Bring the mixture up to a gentle boil. Add 1 cup of the hot cream mixture to the beaten yolks and whisk well. Add the yolk mixture back to the cream mixture and whisk well. Continue to cook for 4 to 6 minutes, or until the mixture can coat the back of a spoon. Remove from the heat, and pour into a glass bowl. Press a piece of plastic wrap to the cream mixture (this will prevent a film from forming) and cool completely. Pour the cream mixture into a frozen ice-cream cylinder and process according to manufacturers directions. In a medium sized bowl, combine the cherries, remaining 1/2 cup sugar, and rum (to taste). Mix well and allow to sit for 10 minutes. To serve, place a couple scoops of ice cream in each serving bowl. Spoon the cherries over the ice cream and serve with the butter cookies.