Preheat the oven to 350 degrees F (175 degrees C). Spray a 9x5-inch loaf pan with cooking spray, or line with parchment paper.
Sprinkle chopped plums with 1 tablespoon of flour in a bowl and toss to coat; set aside.
Beat white sugar, butter, and vanilla extract in a large bowl with an electric mixer until light and fluffy; beat in eggs. Sift 1 ½ cups of flour, salt, and baking soda into a separate bowl; stir into egg mixture, alternating with yogurt until mixture is smooth. Fold in flour-coated plums; pour batter into the prepared loaf pan. Sprinkle brown sugar on top.
Bake in the preheated oven until a toothpick inserted into the center of the loaf comes out clean, 45 to 50 minutes. Remove from oven, and let cool in the pan for 15 minutes before inverting onto a wire rack to cool completely.