Heat the oven to 350°F (175°C). Line a standard 12-cup muffin pan with paper liners, or grease the wells thoroughly.

In a large bowl, beat the softened butter and granulated sugar until lighter in color and creamy, about 2 minutes. Beat in the egg, vanilla, and 1/2 teaspoon rum extract until combined.

In a second bowl, whisk the flour, almond flour, breadcrumbs, baking powder, cardamom, and salt. Add the dry mixture to the butter mixture in two additions, alternating with the milk. Stir just until no dry streaks remain; do not overmix.

Divide the batter evenly among the prepared muffin cups, filling each about two-thirds full. Bake for 18 to 22 minutes, until the tops spring back lightly and a toothpick inserted in the center comes out clean or with a few moist crumbs.

While the cakes bake, combine the water and 2 tablespoons sugar in a small saucepan. Bring to a gentle simmer, stirring until the sugar dissolves. Remove from the heat and stir in 1/4 teaspoon rum extract. Let the syrup cool slightly.

When the cakes come from the oven, poke each one several times with a toothpick. Spoon about 1 teaspoon of syrup over each warm cake. Let the cakes cool in the pan for 10 minutes, then transfer them to a rack and cool completely.

Stir the raspberry jam until smooth. If it is very thick, warm it briefly so it is easy to spoon. In a small bowl, mix the powdered sugar with 1 teaspoon water. Add more water a few drops at a time until the icing is thick enough to hold its shape when piped.

Spoon the icing into a small zip-top bag and snip off a tiny corner, or use a piping bag. Pipe a ring around the top edge of each cooled cake. Spoon about 1 teaspoon raspberry jam into the center of each icing ring. Let the icing set before serving.