Special equipment: an instant-read or candy thermometer; an ice cream maker; a 3-inch cookie cutter
Directions
Bring cream, half-and-half, sage, and zest to a boil in a large heavy saucepan over moderate heat. Remove from heat and steep, covered, 10 minutes.
Whisk together yolks, granulated sugar, and salt in a large bowl. Whisk in half of hot cream, then whisk egg mixture into remaining cream in saucepan. Cook custard over moderate heat, stirring constantly with a wooden spoon, until it coats back of spoon and reaches 170°F on thermometer, about 5 minutes (do not let boil).
Pour custard through a fine sieve into a bowl and cool, stirring occasionally. Chill custard, its surface covered with plastic wrap, until cold, at least 3 hours.
Freeze custard in ice cream maker, then transfer to an airtight container and put in freezer to harden.
Preheat oven to 350°F.
Butter a 13- by 9- by 2-inch metal baking pan. Line bottom with wax or parchment paper, then butter paper.
Sift together flour, cinnamon, allspice, baking soda, and salt. Whisk together brown sugar and eggs in a large bowl, then whisk in oil and pumpkin purée. 3Add flour mixture and whisk just until smooth. Pour batter into baking pan and bake in middle of oven until springy to the touch and a tester inserted in center comes out clean, 25 to 30 minutes.
Cool cake in pan on a rack 10 minutes, then run a knife around edge and invert onto rack. Peel off paper and cool cake completely.
Soak cherries in hot water to cover until softened, about 15 minutes, then drain.
Melt butter in a large skillet over moderate heat. Add sugars, lemon juice, cinnamon, and salt, then cook, stirring until smooth. Add pumpkin and drained cherries and simmer, covered, stirring occasionally, until pumpkin is tender, 8 to 12 minutes.
Cut out 12 (3-inch) rounds from cake with cutter. Halve each round horizontally, then put bottom halves on 12 plates and top with scoops of ice cream. Spoon compote on and around cakes, then tilt tops against ice cream.
