Preheat your oven to 350 degrees F. Lightly butter an 8½-by-4½-inch loaf pan; press and 8½-inch wide strip of parchment paper into bottom of pan so it lines the bottom and long edges, leaving two overhangs on either side.
Toss strawberries with granulated sugar and dump into a fine mesh sieve and set it over a bowl to catch the liquid that drains out. Let sit for 30 minutes, tossing berries once or twice to redistribute the sugar.
Melt butter in the microwave or a small saucepan over low heat; set aside and let cool to lukewarm.
In a bowl, whisk together flour, baking soda, cardamom, and salt.
In another bowl, mash bananas until smooth. Whisk in sugar, brown sugar, eggs, and vanilla until evenly incorporated.
Add banana mixture and lukewarm melted butter to dry ingredients, and mix until just incorporated and no dry ingredients remain (batter will still be slightly lumpy).
By this point your strawberries should have drained off about 3 tablespoons of liquid, which you'll use it to make the glaze later (so don't throw it out!).
Spread the strained strawberries onto a paper towel-lined plate and gently pat dry with another paper towel. Sprinkle with 2 tablespoons of flour and toss to coat; this will keep the strawberries from sticking and clumping together.
Fold floured strawberries into batter until evenly incorporated, then pour batter into prepared loaf pan.
If desired, butter or oil the edge of a bench scraper and use that to imprint a fine line of fat down the center of the loaf (this will help form a nice defined crack as it bakes, but no worries if you skip this step).
Bake for about 55 to 65 minutes or until a toothpick inserted in the deepest part of the bread comes out with moist crumbs but no longer any gooey batter. If using a 9-by-5-inch pan, the loaf will be shallower so the cook time will be slightly less (check after 45 minutes or so).
Remove from oven and place pan on a wire rack to allow bread to cool for 30 minutes, then use the parchment overhang as handles to gently lift the loaf out of the pan and place on a wire rack.
Sift powdered sugar into reserved sugar syrup 1 tablespoon at a time, whisking until smooth and no clumps remain. Use a pastry brush to brush an even layer over the top and sides of the entire loaf (if you have a particularly deep crack, try not to let too much of the syrup pool in there as it may get somewhat soggy).
Let glaze set and loaf cool completely before slicing and serving. Store in an airtight container for up to 1 week.
