Adjust oven rack to the center position and preheat oven to 350 degrees. Spray or oil a nonstick 12-cup fluted cake pan and lightly dust with flour (make sure to get into the crannies)
Toss the cherries in a small bowl with 2 tablespoons flour.
Stir together the remaining 2 3/4 cups flour, sugar, baking powder, salt and baking soda in a large bowl until combined.
Whisk the eggs in a medium bowl. Drizzle in the oil little by little until emulsified. Stir in the milk and extracts.
Pour the egg mixture over the flour mixture, stirring just until the batter is smooth and no flour is visible. Gently fold in the cherries.
Scrape the batter into the pan. Bake 50-55 minutes, until the cake is golden brown and a toothpick inserted in the middle comes out dry. Cool the cake in the pan 20 minutes, then invert onto a cooling rack to cool completely.
Stir the sugar and 2 tablespoons heavy cream in a bowl until smooth. Add more cream if needed by the tablespoon until you have a thick but spreadable consistency. Set the cake on a wire rack over a piece of parchment paper and drizzle the glaze over. Let it set slightly before slicing.
Glazed cake will keep for 2 days at room temperature.
Cake can be made 2 days before glazing - wrap securely in plastic and store at room temperature.
You can decorate the cake with whole cherries and some tender rosemary sprigs to fancy it up a little.