Pears. Preheat oven to 350°F. Line a jelly roll pan or 9 inch by 13 inch pan with parchment paper or non-stick foil. In a medium bowl, stir together the maple syrup, cinnamon and vanilla. Toss pears in maple mixture. Spread pears on the prepared pan so they are in a single layer and not piled on top of each other. Roast pears for 50-60 minutes, tossing them every 15-20 minutes. The pears are done when a fork can easily pierce through them. (For crunchier pears, roast for less time, approximately 30 minutes.) Allow to cool on pan. Once cooled to room temperature, strain pears from the juices, reserving the juices in a separate bowl. Pears can be prepared up to several days in advance. Store the pears and the juices in separate airtight containers in the refrigerator.
Cake. Preheat oven to 350°F. Lightly grease three 8-inch round pans, line each with circles of parchment on the bottom and grease the top of the parchment. In a large bowl, whisk together all the dry ingredients through the dark brown sugar. Whisk in the oil, eggs and vanilla until combined. Whisk in milk and sour cream until combined. Pour into prepared pans and bake 23-27 minutes, until a toothpick inserted comes out clean and the sides are just starting to pull away from the pans. Cool completely in pans.
Maple Cream Cheese Frosting. Using the paddle attachment of the mixer, beat the cream cheese and butter together for a minute or two until fully combined and the mixture is lightly aerated. Add the maple extract and the salt, beating until combined. Stop the mixer and pour in confectioner’s sugar. Start with the mixer on the lowest speed, raising it as the confectioner’s sugar gets incorporated into the mixture. Beat on high for a minute or two.
Assembly. Slide a metal spatula or knife around the edges of each cake pan and remove the cakes from the pans. If necessary, carefully cut the tops of the the cakes off to even them out. With a pastry brush, brush on the tops and bottoms of each cake the reserved juices from the roasted pears. Fill a pastry bag with frosting and pipe a continuous ring on top of one of the cakes, completely covering the top. With a metal spatula, spread the frosting evenly over the top. Spoon no more than half of roasted pears onto the frosting. The pears should be spread to create one layer on top of the frosting and not be piled up on each other. Place another cake on top and repeat with the piping and pear layer. Place the last layer on top and frost the top and sides as desired.
Store cake in refrigerator, allowing it to come to room temperature approximately 30 minutes before serving.