Make tea bread: Preheat oven to 350°F. Butter and flour a 9- by 5-inch loaf pan.
Stir together 1 1/2 cups sifted cake flour, 1 tsp. baking powder and 1/4 tsp. salt. Mix 1/2 tsp. vanilla extract with 1/2 cup milk.
Using electric mixer, cream 8 Tbsp. unsalted butter, 3/4 cup sugar and 1 Tbsp. lemon zest on medium-high speed until fluffy. Add 2 large eggs, one at a time, mixing well after each addition. Mix in dry ingredients alternately with milk, beginning and ending with flour mixture. Stir in 1 1/2 Tbsp. lemon juice and 2 Tbsp. poppy seeds.
Pour batter into pan and bake in center of oven until a tester inserted in center of loaf comes out clean, 50 minutes.
Make syrup: Mix 1/2 cup sugar and 1/4 cup lemon juice in a saucepan over low heat; stir until sugar dissolves.
Let loaf cool in pan on a wire rack for 10 minutes before turning loaf out onto rack. Poke holes in loaf and gradually spoon warm syrup over, adding more as syrup is absorbed. Cool completely.