Put the sugar in a large bowl and sift in 1 cup of the flour and the salt; whisk briefly to combine. Slowly add the beaten eggs while stirring with a wooden spoon, then continue to stir forcefully until the mixture becomes a thick, creamy paste, about 4 to 5 minutes.

Fill another large bowl with cold water. Melt ½ cup (1 stick) of the butter in a small saucepan over medium heat and cook, stirring occasionally, until it just begins to turn brown, about 5 minutes. Remove from the heat and submerge the bottom of the pan in the cold water, stirring the butter constantly until it reaches room temperature, about 5-6 minutes. Add the butter to the flour mixture, and stir with the wooden spoon until the mixture is glossy and has the consistency of pulled taffy, 2 to 3 minutes. Add the vanilla bean seeds, vanilla extract, lemon zest and juice, and rose water, if using, and stir to combine. Cover the bowl with plastic wrap and let rest in the refrigerator for at least 2 hours and up to overnight.

Heat the oven to 375°F. Melt the remaining 2 tablespoons (¼ stick) of the butter in a small saucepan over medium-low heat and cook, stirring, until lightly browned, about 3 minutes. Combine the brown butter with the remaining 1 tablespoon flour in a small bowl.

Using a pastry brush, coat the cups of a 12-madeleine pan with a thin film of the butter-flour mixture; don’t let it pool in the bottom of each cup. Spoon a ping-pong-ball-sized scoop of dough into the center of each cup. Use your fingers to gently press each ball of dough into a rough oval. Bake until the centers of the madeleines are puffed and the edges just browned, 13 to 15 minutes—watch carefully toward the end as they can burn quickly. Let the madeleines cool for 5 minutes in the pan, then transfer them to wire racks to finish cooling.

Repeat Step 4 with the remaining butter-flour mixture (you may need to stir in a few drops of hot water if it has congealed) and the remaining dough. Serve plain or dusted with powdered sugar. (Store madeleines in an airtight container at room temperature for up to a few days.)