Step 1Heat the oven to 350 degrees. Generously butter a 9-by-1 1/2 -inch round cake pan.
Step 2Sift the flour, salt and baking powder together. Stir in the cardamom and set aside.
Step 3Melt one-fourth cup butter in a small saucepan over medium heat. Add the brown sugar and stir for 2 to 3 minutes, until the sugar has melted and combined with the butter. Pour the mixture into the prepared cake pan, spreading it to reach the sides.
Step 4Peel the pears, cut in half and remove the core and stem. Cut each half crosswise into one-fourth-inch-thick slices. Arrange the pear slices in a slightly overlapping circle around the cake pan, starting at the outer rim. Finish with several slices in the center. Sprinkle the pears with the lemon juice. Cover with plastic wrap and set aside.
Step 5Beat the remaining one-half cup butter in the bowl of an electric mixer until soft and fluffy. Add the sugar and beat until smooth. Add the eggs one at a time, beating after each addition. Beat in the vanilla, scraping down the sides of the bowl when needed. Alternately add the flour mixture and the milk, beating after each addition just until combined.
Step 6Gently spoon the cake batter on top of the pears, smoothing out to the edge of the pan and making sure the cake batter fills in around the pears.
Step 7Bake until the top is a deep golden brown and a skewer inserted in the center comes out clean, about 40 minutes. Place the cake on a rack to cool for 5 minutes in the pan.
Step 8Run a small spatula or knife around the edge of the pan and invert onto a cake plate, leaving the pan on the cake for 10 minutes. Carefully remove the pan. Serve warm or at room temperature.