To prepare pears, working with 1 pear at a time, hold pear, stem side down, in 1 hand. Make 3 or 4 quick cuts into pear from the bottom, using a melon baller (do not remove stem). If necessary, cut about 1/4 inch from base of each pear so they will sit flat when served.
Combine 3 cups water and next 5 ingredients (through wine) in a large stockpot over medium heat; bring to a simmer. Cook 6 minutes or until sugar dissolves, stirring occasionally. Add pears. Using tongs, place a small clean plate on top of pears to weigh them down. Return to a simmer; cook 15 minutes or until tender. Remove pot from heat; cool mixture to room temperature. Cover and chill 4 hours or up to overnight (do not remove plate).
To prepare cream, melt ice cream in a small heavy saucepan over medium-low heat. Remove from heat.
Heat a small skillet over medium-high heat. Add cardamom; cook 2 minutes or until fragrant, shaking pan frequently. Stir cardamom into melted ice cream; cook over medium-low heat 5 minutes, stirring occasionally. Remove from heat. Strain mixture through a fine sieve over a bowl; discard solids. Cool cream to room temperature; cover and chill.
Remove the plate from chilled pears. Remove pears from liquid with a slotted spoon; discard liquid. Spoon about 2 1/2 tablespoons cream onto bottom of each of 6 small dessert plates or shallow bowls; top each serving with 1 pear. Garnish with mint sprigs and ground cardamom, if desired.