Preheat oven to 350F (175C). Line an 8x8-inch baking pan with parchment paper, leaving 1-inch overhang on all four sides.
In a large bowl, combine flour, oats, brown sugar, baking powder, and salt. Add cold butter cubes and work in with fingertips until the mixture resembles coarse crumbs with pea-sized butter pieces. Do not overmix.
Press two-thirds of the crumb mixture firmly into the pan. Reserve remaining third for topping. Bake for 12 to 14 minutes until edges turn golden. Remove from oven but do not turn it off.
Whisk together sweetened condensed milk, fresh lemon juice, lemon zest, and vanilla (if using) until smooth and slightly thickened, about 60 seconds. Do not overbeat.
Pour lemon filling evenly over the warm crust. Sprinkle reserved crumb topping evenly over the filling.
Bake at 350F for 20 to 25 minutes until edges are golden and center has a gel-like set (not a liquid ripple) when pan is gently shaken.
Cool completely at room temperature, about 1 hour. Then refrigerate for a minimum of 4 hours. Overnight is strongly preferred before cutting.
Lift the chilled slab from the pan using the parchment overhang. Use a sharp knife dipped in hot water (wiped dry) to cut into 16 squares with a single downward press. Clean knife between every cut.
