For the pears: Lightly butter a 9-inch round cake pan, line the bottom with parchment paper, and butter the paper. Put the butter, brown sugar, and lemon zest and juice in a large skillet. Cook over medium heat, stirring, until melted. Add the pears, cover, and simmer, turning once, until the pears are soft and juicy, about 4 minutes. Pour the pears and their syrup into a strainer set over a saucepan. Set the syrup and 1 pear quarter aside. Arrange remaining pears in the prepared pan, stem ends toward center, rounded sides down, and 1/4-inch from the side. Chop reserved pear and use just enough to fill center.
For the cake: Set a rack in the middle of the oven and preheat to 350 degrees F.
Scald milk in a small saucepan; keep warm. Whisk flour with almonds, baking soda, baking powder, salt, cinnamon, and cloves. Beat butter with an electric mixer at medium speed until smooth, about 2 minutes. Increase speed to medium-high and gradually add granulated sugar; continue beating until mixture is light and fluffy, about 4 minutes, scraping down sides of bowl. Add cocoa powder and vanilla and beat 1 minute. While mixing at medium-low speed, add eggs 1 at a time, beating well after each addition.
While mixing slowly, add the flour mixture in 4 additions; turn off the mixer and fold in the hot milk with a rubber spatula. Spread the batter over the pears, taking care not to disturb them. Tap the pan lightly on the counter to settle the batter. Bake until a toothpick inserted in the center comes out clean, about 50 minutes. Cool cake in the pan on a rack for 20 minutes, then invert the cake onto a serving plate. Cool briefly, carefully peel the parchment paper from the top, and then cool completely.
To finish the cake: Boil the reserved pear syrup until slightly thick and lightly brush the top of the cake with syrup. Whisk a few tablespoons of cocoa powder with an equal amount of confectioners' sugar, and sift over the cake; serve.