Heat oven to 350°F. Grease a 9×13-inch pan and line the bottom with parchment.
Zest and juice limes. Measure out zest and divide the juice for batter and glaze.
Whisk flour, granulated sugar, baking powder, and salt in a large bowl.
Whisk eggs, sour cream, buttermilk, melted butter, oil, vanilla, and 1/3 cup lime juice in a separate bowl until smooth.
Pour wet ingredients into dry. Stir just until combined, then fold in lime zest.
Spread batter in the pan and smooth the top. Tap the pan once to release bubbles.
Bake 28–32 minutes, until a toothpick comes out clean. Cool in pan 15 minutes.
Whisk powdered sugar, 3 tbsp lime juice, melted butter, and a pinch of salt until pourable.
Poke the warm cake lightly with a skewer, then drizzle on the glaze. Let set 20 minutes before slicing.