Preparation of Baking Equipment and Preheating: Position a rack in the middle of the oven. Preheat the oven to 350 degrees Fahrenheit (177 degrees Celsius). Line an 8x8-inch baking pan with two sheets of parchment paper, placed perpendicularly. Ensure the sheets extend at least one inch over all four sides to facilitate easy removal of the bars.
Melting the Butter: Cut the butter into eight pieces and place them in a large, microwave-safe bowl. Melt the butter in the microwave using 20-second intervals, checking frequently, until mostly melted (approximately 40 to 60 seconds total). Alternatively, the butter may be melted on the stovetop and then transferred to a bowl.
Preparing the Crumb Mixture: To the melted butter, add the brown sugar, baking powder, and 1/2 teaspoon of the kosher salt. Stir to blend thoroughly. Incorporate the oats and all-purpose flour, mixing with a flexible spatula until the mixture is uniform and no dry spots remain. Reserve one cup of this crumb mixture for later use as the topping.
Baking the Base Layer: Transfer the remaining crumb mixture to the prepared baking pan. Press the mixture into a compact, even layer using the back of a spoon or your hands. Bake in the preheated oven until it is fragrant and lightly browned, about 12 to 14 minutes. Remove the pan from the oven and set aside.
Mixing the Lemon Filling: Using the same bowl (no need to wash it), finely grate the zest of the lemons to yield about 2 tablespoons of zest. Juice the zested lemons to obtain 1/2 cup of lemon juice. Combine the lemon zest, lemon juice, sweetened condensed milk, and the remaining 1/4 teaspoon kosher salt in the bowl. Whisk until the mixture is smooth.
Assembling and Baking the Filled Bars: Pour the lemon filling over the hot, pre-baked crust. Evenly sprinkle the reserved cup of crumb mixture over the filling. Return the pan to the oven and bake until the edges are well-set and lightly browned, about 22 to 24 minutes.
Cooling and Cutting the Bars: Transfer the baking pan to a wire rack and allow the bars to cool completely, approximately 1 1/2 hours. If a firmer filling is desired, the cooled bars can be refrigerated for 30 minutes. Using the overhanging parchment as handles, lift the slab of bars from the pan and place it on a cutting board. Cut into 12 equal pieces.