Preheat oven to 350°F (175°C). Grease a 9x5 inch loaf pan, line with a parchment paper sling, and set aside.
Zest all 3 lemons to get about 3 tablespoons of zest, then cut and juice them. Set the zest and juice aside separately.
In a large bowl, whisk together the flour, poppy seeds, baking powder, baking soda, and salt for 30 seconds until combined. Set aside.
In a second large bowl, beat the softened butter and granulated sugar with an electric mixer on medium-high speed for 3–4 minutes, scraping the bowl halfway through, until the mixture is pale, fluffy, and light.
Add the eggs one at a time, beating 20 seconds after each addition, then mix in the lemon zest, ¼ cup lemon juice, and vanilla extract on low speed until just combined.
Add the sour cream and mix on low speed or fold in with a spatula until fully incorporated into the batter.
Add the dry ingredient mixture in two batches, folding gently with a rubber spatula just until the last streak of flour disappears — do not overmix.
Pour batter into the prepared pan, smooth the top, and tap the pan twice on the counter. Bake on the center rack for 55–60 minutes, tenting loosely with foil during the last 10–15 minutes if the top is browning too quickly, until a toothpick inserted in the center comes out clean.
Cool the cake in the pan on a rack for 15 minutes, then lift it out using the parchment sling and let it cool on the rack for another 10–15 minutes until just warm.
Whisk together the powdered sugar and 3 tablespoons of fresh lemon juice in a small bowl until completely smooth; add the remaining tablespoon of juice if the glaze is too thick to pour.
Place the warm cake over parchment to catch drips, pour the glaze evenly over the top, and let it set for 10–15 minutes before slicing with a serrated knife.