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1
Preheat the oven to 350F.
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2
Butter an 8-by-8-by-2-inch cake pan, line it with parchment paper, and butter parchment.
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3
Using a 3-inch cookie cutter, cut out rounds from centers of four of the pineapple slices.
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4
Using a 1 1/2-inch cookie cutter, cut out the centers of each of the four rounds, and discard; set rings aside.
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5
Finely chop enough of the remaining pineapple to yield 1 cup (reserve remainder for another use).
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6
Place chopped fruit in a fine sieve set over a bowl to drain.
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7
Cut the mango lengthwise into 1/4-inch-thick slices.
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8
Using the 1 1/2-inch cookie cutter, cut out four rounds from mango slices; set aside.
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9
Finely chop enough of the remaining mango to yield 1/2 cup; add to the pineapple in the sieve.
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10
In the bowl of an electric mixer fitted with the paddle attachment, beat 2 tablespoons butter with the brown sugar until light and fluffy, 2 to 3 minutes.
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11
Using a small offset spatula, evenly spread butter mixture in the bottom of the prepared pan; set aside.
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12
In a medium bowl, sift together the flour, salt, baking soda, and baking powder; set aside.
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13
Place a reserved pineapple ring in each corner of the prepared pan.
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14
Place a reserved mango round in the center of each ring.
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15
In the bowl of a mixer fitted with the paddle attachment, beat remaining 1 stick butter with granulated sugar on medium speed until light and fluffy, 2 to 3 minutes.
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16
Add eggs, one at a time, beating well after each addition; scrape down the sides of the bowl as needed.
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17
Beat in the vanilla.
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18
With the mixer on lowest speed, add flour mixture in two parts, alternating with sour cream and beginning and ending with flour.
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19
Transfer the chopped pineapple-mango mixture to cake pan; discard juice.
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20
Using a small offset spatula, carefully spread fruit in an even layer on top of cut-out fruit, making sure to fill all the empty spaces, including any gaps in the corners.
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21
Using the offset spatula, spread batter evenly over fruit.
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22
Bake, rotating pan halfway through, until cake is golden brown and a cake tester inserted in the center comes out clean, about 50 minutes.
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23
Transfer pan to a wire rack to cool 30 minutes.
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24
Invert onto a cake plate; peel off the parchment paper.
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25
Serve cake slightly warm or at room temperature.
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26
Pineapple cores are removed with a round cookie cutter; the same cutter is used to make mango rounds that will fill in the holes in the pineapple rings.
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27
Some of the remaining fruit is finely chopped and drained.
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28
After pineapple rings and mango rounds are arranged on the bottom of the pan, the drained, chopped fruit is spread on top, before the cake batter.
