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For the dough: Place the butter, sugar, and salt in a large mixing bowl. Bring the milk to a simmer on the stove or in the microwave and pour over the ingredients in the bowl. Stir briefly and allow to cool to lukewarm.
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When the mixture has cooled, stir in the yeast, egg, and vanilla. Add the flour and mix until a soft dough forms. Knead the dough for 6 minutes in a mixer or 8 minutes by hand. Cover and let rise until doubled, about 45 minutes to 1 hour.
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For the filling: In a medium bowl, whisk together the egg white, sugar, and salt. Stir in the toasted almond flour and all-purpose flour.
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Add the melted butter, almond extract, and 2 teaspoons of milk. The filling should be quite thick but spreadable, about the consistency of peanut butter. Add a little more milk if needed to achieve the desired consistency.
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To assemble: Roll out the dough on a greased surface to a rectangle 24"wide and 10" tall. Spread the filling over the dough in a thin layer, leaving 1" of both long edges uncovered. Roll up the dough from one long edge to the other, pinching it together to seal it.
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Transfer the dough to a piece of parchment, seam side down, and form into a circle, tucking one end inside the other. Pinch this seam together also. You should have a circle in the center roughly 4" in diameter.
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Slice the dough from the outside edge three-quarters of the way through, leaving the center intact. Repeat in 1 1/2" increments all the way around the dough (spin the piece of parchment with the dough on it to make this easier).
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Turn the slices on their sides, pulling every other one to the inside of the circle. Cover with greased plastic and let rise once more until puffy, about 45 minutes.
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Preheat the oven to 350°F.
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When the dough has risen, brush it all over with the egg wash and bake for 24 to 26 minutes, until golden brown and the center reads 190°F when measured with a digital thermometer. Remove from the oven and let cool on a rack.
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For the glaze: Combine all the ingredients except for the almond slices. Drizzle over the slightly warm coffee cake. Sprinkle with toasted almonds.
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Store, well-wrapped, for 3 days at room temperature, or freeze for up to 3 months.