Dissolve the syrup sugar. Combine the syrup water and sugar in a medium saucepan. Heat while stirring until the sugar has dissolved completely.
Simmer the light agda. Bring to a boil, then simmer gently over medium heat for 5 minutes. Add the lemon juice and slices and simmer for 3 more minutes. Keep the syrup pourable; do not reduce it to a thick consistency.
Cool completely. Pour the agda into a heatproof bowl and let it cool completely to room temperature. A cold-water bath around the bowl can help it cool sooner. Keep bath water out of the syrup and remove the lemon slices after cooling.
Prepare the oven and pan. Heat a conventional oven to 355°F (180°C), with a rack in the middle. Set out a deep metal pan about 15 × 11 inches (38 × 28 cm). No greasing is needed for this butter-rich dough. The pan must be deep enough to hold the syrup.
Cream butter and sugar. Beat the soft butter and dough sugar with a hand mixer for 3–4 minutes, until creamy. The butter should be pliable, not melted.
Add the egg, yolk, and yogurt. Beat in the whole egg completely, then beat in the yolk. Briefly mix in the plain yogurt until combined.
Combine the dry ingredients. In a separate bowl, mix the flour, baking powder, and salt thoroughly so the leavener is evenly distributed.
Make a soft dough. Add the dry mixture to the butter mixture in portions, using low mixer speed at first. Finish briefly with your hands only until the dough is smooth and combined. Do not knead it like bread or keep working it after it comes together.
Rest for 10 minutes. Let the dough rest for 10 minutes. It should be soft, shapeable, and barely sticky rather than dry or stiff. Do not add more flour just to make it feel firm.
Portion evenly. Divide into equal portions of about 31 g each, making 30 cookies per standard batch. For larger batches, increase the number of pieces rather than their size.
Shape and imprint. Roll each portion into a ball, then an oval about 2⅜ inches (6 cm) long. Gently press the top against the coarse face of a clean grater to make the pattern, lengthening it slightly without forcing dough through the holes. Keep the oval plump rather than flattening it into a thin disk.
Arrange in the pan. Place the cookies pattern-side up in the deep metal pan, leaving about ¾ inch (2 cm) between them. Use additional comparable pans for larger batches.
Bake until golden. Bake on the middle rack for about 25–30 minutes, until evenly golden brown without becoming dark or dry. Baking time depends on the pan and oven; retain the same cookie size when scaling.
Pour cool syrup over hot cookies. Remove the pan from the oven and wait 2 minutes. Slowly pour the completely cooled agda between the still-hot cookies first, then over their tops. Use a metal pan to avoid the thermal-shock risk of glass or ceramic.
Baste after 20 minutes. After about 20 minutes, spoon some syrup from the pan over any tops that are not fully moistened. Handle the cookies gently and do not press them down.
Soak before serving. Let stand at room temperature for at least 4 hours, uncovered for the first hour. Cover loosely only after completely cool. An overnight rest gives more evenly soaked cookies; refrigerate after the soaking period. Serve with some of the pan syrup.