Preheat the oven to 375 degrees F (190 degrees C).

Place self-rising flour in a large bowl. Cut in butter using a pastry blender until mixture resembles coarse crumbs. Add poppy seeds, sugar, and lemon zest; mix to combine. Make a well in the center and pour in egg, lemon juice, and milk. Mix with a fork until dough starts to come together. Scoop up with one hand until a shaggy dough forms.

Pour dough out onto your work surface. Use your hands and a bench scraper to gather and press dough into a neat rectangle about 1- to 1-1/2-inches tall. Cut into 4 squares and then diagonally into 8 triangles. Space scones evenly onto a silicone-lined baking sheet.

Bake in the preheated oven until bottoms are golden brown, about 25 minutes. Let cool in pan for 5 minutes, then transfer onto a wire rack to cool completely, about 30 minutes.

Mash strawberries finely to resemble a puree. Whisk in powdered sugar in batches to make a thick, spreadable glaze. Spread glaze over the cooled scones. Let scones cool on the rack until glaze hardens, about 20 minutes.