For the streusel: Preheat the oven to 325 degrees F.
Stir together the pecans, flour, brown sugar, granulated sugar, cinnamon and butter in a large bowl until wet, sandy and combined. Spread 3/4 cup on a baking sheet (set the remaining aside for the cake). Bake until the streusel is set and crispy, about 15 minutes. Break up the pieces with a spatula and let cool completely in the pan, about 30 minutes. Break up the pieces into crumbs with your hands and set aside.
For the cake: Grease a 9-by-13-inch baking sheet with butter. Whisk the flour, baking powder, baking soda and salt in a medium bowl.
Beat the butter and granulated sugar in a large bowl with an electric mixer on medium-high speed until light and fluffy, about 3 minutes. Add the eggs, one at a time, beating well after each addition. Beat in the vanilla. Reduce the speed to low and add the flour mixture in 3 batches, alternating with the sour cream; begin and end with the flour and beat until just incorporated.
Pour the batter into the prepared pan (it will be very thick) and smooth with a silicone spatula. Press 15 mini doughnuts or 6 regular doughnuts into the batter and sprinkle over the reserved un-baked streusel topping. Bake until a toothpick inserted into the center of the cake comes out clean, 60 to 65 minutes. Run a knife around the edge of the pan to loosen any streusel that has baked on. Let cool in the pan for 1 hour, then invert onto a serving platter to cool completely, about 2 hours more.
For the frosting: Meanwhile, beat the cream cheese and butter in a medium bowl with an electric mixer on medium speed until light and fluffy. Slowly add 1 1/2 cups confectioners' sugar, 1/2 cup at a time, until well combined. Add the vanilla, beating until well combined.
Assemble the cake: To make a glaze, mix the remaining 1/2 cup confectioners' sugar in a small bowl with 1 tablespoon water until smooth. Use a serrated knife to trim the edges from the short sides of the cake to make sharp corners. Cut the cake in half crosswise.
Put a cake half on a cake stand or serving platter. Reserve 1/4 cup frosting and spread the remaining frosting on the top of the cake, then place the other piece of cake on top. Cut the remaining 4 mini doughnuts or 3 regular doughnuts in half and spread some of the reserved frosting on the cut ends. Put the doughnuts on the top of the cake around the edge, cut-side down. Sprinkle the reserved streusel on the center of the cake and drizzle with the glaze. Refrigerate to set, about 2 hours. Bring to room temperature, about 1 hour, before serving.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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