Heat a conventional oven to 350°F (175°C) with a rack in the lower-middle position. Trace the base of a solid 9 x 2-inch metal cake pan onto parchment and cut out the circle. Lightly grease the pan sides, fit the parchment snugly over the bottom, and set the pan on a dry work surface. Avoid a springform pan because the warm butter and sugar topping can leak through its seam.
Place the 170 g butter in a small light-colored saucepan and melt it over medium heat. Continue cooking for 5 to 7 minutes, swirling frequently as the butter foams and crackles. Remove it from the heat when the foam subsides, the milk solids are deep golden brown, and the aroma is distinctly nutty. Immediately scrape the butter and every browned particle into a heatproof bowl. Cool for about 15 minutes, until fluid but only barely warm.
Pour the 28 g melted butter over the parchment-lined pan and tilt the pan to coat the base. Sprinkle the brown sugar evenly over the butter, breaking up firm lumps with your fingertips. Arrange the plum wedges skin-side down in tight, slightly overlapping concentric circles, filling the center with smaller pieces. Keep the fruit in one even layer rather than stacking it, which helps the cake bake level and limits pools of uncooked juice.
Set a fine-mesh sieve over a medium bowl and add the all-purpose flour, almond flour, baking powder, and salt. Sift the mixture, then whisk for about 20 seconds to distribute the almond flour and leavener evenly. Break apart any almond flour clumps that remain. This even distribution matters because concentrated pockets of almond flour can turn pasty beside the fruit, while uneven baking powder can leave tunnels in the crumb.
In a large bowl, whisk the cooled brown butter and granulated sugar for 45 seconds; the mixture will look glossy and slightly grainy rather than creamed. Whisk in the eggs one at a time, mixing thoroughly after each, then add the vanilla. Whisk in the sour cream until smooth, followed by the milk. The mixture should be cohesive and satiny. If small butter flecks appear, let the bowl stand in a warm spot for 2 minutes and whisk again.
Add the dry mixture to the wet ingredients in two additions, folding with a flexible spatula just until no flour streaks remain. The batter should be thick, smooth, and able to fall slowly from the spatula. Spoon it gently over the plums, then spread it to the edge without disturbing the pattern. Bake for 47 to 52 minutes, until the center springs back, the top is golden, and a tester inserted in the center returns with moist crumbs but no wet batter.
Transfer the pan to a wire rack and leave the cake undisturbed for exactly 15 minutes. Run a thin offset spatula around the side without cutting toward the fruit. Center a flat serving plate over the pan, protect your hands with dry oven mitts, and invert both in one confident motion. Wait 30 seconds before lifting the pan, then peel away the parchment. Reposition any attached plum gently and cool for at least 1 hour before slicing with a sharp knife.