Preheat your oven to 350°F (177°C). Line an 8-inch square baking pan with parchment paper, leaving an overhang on the sides to facilitate lifting the bars out later.
Prepare the Crust and Topping: In a mixing bowl, beat the softened butter and brown sugar together using a handheld or stand mixer with a paddle attachment on medium-high speed for about 1 minute, until well combined. Scrape down the sides and bottom of the bowl with a rubber spatula. Add the vanilla extract and mix again on medium-high speed until fully incorporated, approximately 30 seconds. Gradually add the flour, baking powder, salt, and oats, and mix on medium speed until everything is combined. Scrape the sides and bottom of the bowl and mix once more if necessary. The mixture will be dry and crumbly, yielding between 3 to 3.5 cups of the crumble mixture.
Firmly press a little more than half of the crumble mixture into the bottom of the prepared baking pan, reserving the rest for the topping. Ensure that there is slightly more mixture used for the base than for the topping. Bake the crust for 12 minutes.
Make the Lemon Filling: In the meantime, whisk together the sweetened condensed milk, freshly squeezed lemon juice, and lemon zest in a bowl until thick and creamy.
Once the crust is finished pre-baking, pour and evenly spread the lemon filling over the hot crust. Sprinkle the reserved crumble mixture over the lemon layer.
Return the baking pan to the oven and bake for an additional 22–25 minutes, or until the edges are very lightly browned and the bars appear set. Be careful not to over-bake. Remove from the oven and place the pan on a wire rack to cool completely.
Once cooled, use the parchment paper overhang to lift the bars out of the pan. Cut into squares.
Store any leftover bars at room temperature for up to 2 days or refrigerate for up to 1 week.