Preheat oven to 350° and line a 9-inch round cake pan with parchment or coat with cooking spray.

Melt 3 tablespoons butter in a small pan over medium heat; add maple syrup and brown sugar and cook, stirring, until sugar dissolves. Bring to a boil and cook for another 2 to 3 minutes until it has reduced slightly. Remove from the heat and let cool while preparing the pears.

Pour the maple mixture into the prepared 9-inch baking pan and arrange pear slices over the top.

In a large bowl, beat granulated sugar and remaining ½ cup butter until fluffy; 3 to 4 minutes. Add eggs, one at a time, until fully incorporated, and then stir in the vanilla.

In a separate bowl, whisk together flour, baking powder, salt, baking soda, cinnamon, and cardamom. With the mixer on low, gradually add flour mixture to butter mixture alternating with milk, beginning and ending with flour mixture. Fold in sour cream and carefully spread the batter over the pears.

Bake until the top is golden brown and the edges pull away slightly from the pan; 45 to 50 minutes. Or until a wooden pick inserted in the center comes out clean. Let cool in pan for 5 minutes. Run a knife around the edges of the cake and then carefully invert onto a plate, and let cool completely. Cover and store at room temperature for up to 3 days.