Heat oven to 350° F. I put a round of parchment paper on the bottom of a 9-inch springform pan, but didn't prepare the pan otherwise.
Slice your pears so they're ready to go.
Brown the butter: Cut the stick of butter into smaller pieces and add to a medium skillet. Heat on medium until it melts, then continue to heat and stir as it starts to foam, then bubble and crackle, and finally quiet down and the bottom will get dark quickly. Pull before it burns! Set aside to cool slightly.
In a large mixing bowl, whisk together the flour, cornmeal, baking powder, baking soda, salt, 1 cup of the sugar, cinnamon, ginger, and cardamom.
Add the buttermilk, eggs, melted butter, lemon zest, vanilla, and almond extract. Gently fold in the pears.
Transfer the batter to the prepared pan. Bake until a toothpick inserted in the center comes out clean. Use 45 to 55 minutes as a starting place, but mine has been more like 55 to 60 because my pears were very ripe so were adding some extra moisture.
While the cake is baking, stir together the other 1/4 cup of sugar and the 1/4 cup water in a small pot over medium heat. Cook, stirring, until the sugar is dissolved. Remove from heat, stir in almond extract, and set aside. Honestly you could make half this amount and still have some left over.
When the cake comes out of the oven, brush the top with the simple syrup. Cool in the pan for 15 minutes, then transfer to a wire rack to cool.
Serve warm or at room temperature, with whipped cream if desired.