Preheat the oven to 375 degrees F. For the cake: Lightly coat a 9-by-13-inch baking dish with nonstick spray and set aside. Whisk the flour, baking powder, baking soda, salt and pumpkin pie spice together in a large bowl and set aside. In a medium bowl, whisk the eggs and sugars together. Stir in the milk and vanilla, and then add the oil. Whisk in the pumpkin puree and pour the liquid mixture over the dry mixture. Use a wooden spoon to stir the batter together, and then switch to a whisk and give the batter a good 3 or 4 beats before pouring it into the prepared baking dish. Use a rubber spatula to smooth out the batter and place it in the oven until the cake is golden and springs back to light pressure, 25 to 28 minutes. Remove from the oven and set aside to cool completely. Meanwhile, place the pecans on a rimmed baking sheet and toast them until fragrant, about 5 minutes. Transfer the nuts to a large plate to cool, then transfer to a cutting board and finely chop. Place the nuts in a medium bowl and set aside. Divide the cake into quarters and crumble it into a large bowl. Once all of the cake is crumbled, drag 2 forks through the cake in opposite directions to crumble it into a very fine crumb (you can also use a food processor to pulse the cake into fine crumbs). Set aside.
For the frosting: Line a rimmed baking sheet with parchment or waxed paper and set aside. Using a stand mixer fitted with a whisk attachment, beat the cream cheese and butter together on medium speed until airy, about 3 minutes. Add the confectioners' sugar and beat until combined. Increase the speed to medium-high and beat until fluffy, about 5 minutes. Stir the cream cheese frosting into the cake crumbs and, once the mixture looks like wet graham crackers crumbs, roll it into ping pong-size balls. Place the cake balls on the prepared baking sheet and freeze for 1 hour. Place the chocolate in a medium microwave-safe bowl and melt at 50-percent power, stirring every 30 seconds until the chocolate is completely melted, 2 to 2 1/2 minutes (or melt the chocolate over a double boiler, stirring often, until the chocolate is melted). Remove the cake balls from the freezer and insert a lollypop stick or small appetizer-length skewer into the top of each one (stick it in far enough so it is more than halfway through but doesn't puncture the other side of the ball). Dip each ball into the white chocolate, using a spoon to help coat the underside. Tap the stick against the side of the bowl a few times and swirl the stick to encourage the excess chocolate to drip back into the bowl. Dip the top of the ball into the nuts, and then place the cake pop back on the parchment paper-lined baking sheet, nut-side down (so the stick points straight up). Repeat with the remaining cake balls. Let the cake pops stand at room temperature for 10 minutes before serving. Serve that day or refrigerate for up to 3 days and then serve.