Prep the lemons: Wash the Meyer lemons extremely well. Cut them into chunks, and meticulously remove all the seeds.
Puree the citrus: Place the lemon chunks (flesh, pith, and peel) into a food processor. Pulse and blend until the mixture forms a smooth, thick, marmalade-like paste.
Cream the butter: In a large mixing bowl, beat the softened room-temperature butter and granulated sugar together on medium-high speed until pale and fluffy (about 3-5 minutes).
Add the eggs: Beat in the eggs one at a time, ensuring each egg is fully incorporated. Then gently blend in the prepared Meyer lemon puree.
Combine dry ingredients: In a separate bowl, whisk together the all-purpose flour, baking powder, and salt. Gently fold this dry mixture into the wet ingredients using a spatula, mixing just until no streaks of dry flour remain.
Bake the cake: Pour the batter into a greased and parchment-lined 8-inch or 9-inch round cake pan. Bake at 350°F (175°C) for 40-45 minutes, or until a toothpick inserted into the center comes out clean. Let cool before slicing.