Make the batter: Put the milk, yeast, and sugar in the bowl of a stand mixer fitted with the paddle attachment and mix on medium speed until thoroughly combined, about 2 minutes. (You can also make the batter by hand using a whisk.) Reduce the mixer speed to low and add the eggs one at a time, waiting until the first egg is well incorporated before adding the next. Add the flour and salt, and mix until the batter is smooth, scraping down the sides of the mixer bowl as needed.

Remove the bowl from the mixer stand, cover with a towel, and let the batter rise in a warm place until double in volume, about 1 hour. After the batter has risen, transfer it to a large pitcher and set aside.

Line a baking sheet with paper towels.

Peel and core the apple, and cut it cross-wise into ⅛-inch slices. Cut the apple slices into small squares, about ½-inch; they don’t need to be perfectly uniform. Put the apples and ¼ cup water in a small saucepan, bring to a simmer over medium heat, and cook, partially covered, until the apples are tender, about 15 minutes. Using a slotted spoon, transfer the apples to the prepared baking sheet to drain and let cool. 

Melt the butter in a small saucepan over low heat. Remove from the heat and let it sit for a few minutes, allowing the milk solids and water to settle at the bottom. Spoon or pour off the clear butterfat and set aside; discard the milky layer.

Line a baking sheet or tray with paper towels. Heat the aebleskiver pan over medium heat. When the pan is quite hot after about 2-3 minutes, pour about 1 teaspoon of clarified butter into each hollow of the pan, then pour 1 generous tablespoon of the batter into each hollow of the pan making sure the batter reaches the rim. 

Add a few pieces of apple to the center of each doughnut, placing an almond in the center of one. After 1–2 minutes, the edges should be golden brown and most of the batter set, though the centers may still be loose. Insert a knitting needle or wooden skewer into the side of a doughnut and carefully flip it to cook the other side. Repeat with the remaining doughnuts. Continue cooking, turning occasionally, until all sides are browned and the doughnuts are cooked through, about 7 minutes. Using the knitting needle, transfer the fried doughnuts to the paper towel–lined baking sheet.

When all the doughnuts are fried, dust them generously with powdered sugar. Serve with your favorite berry jam alongside.