Grease and line a 9x5 inch (23x13cm) loaf pan with parchment paper, leaving an overhang so the loaf can be easily lifted out once baked.
In the bowl of a stand mixer, cream together the softened butter, vegetable oil, and sugar on medium-high speed for around 4 minutes, until pale in colour and fluffy. Scrape down the sides of the bowl as needed. 115 g unsalted butter, 30 g vegetable oil, 200 g granulated sugar
Add the eggs one at a time, mixing well after each addition, followed by the vanilla. 3 large eggs, 1 tsp vanilla paste
Whisk in the sourdough discard and yoghurt until smooth. If the mixture looks split or grainy at this stage, blend briefly with an immersion blender to bring it back into a smooth emulsion. 100 g sourdough discard, 120 g natural yoghurt
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. 220 g all-purpose flour, 1 tsp baking powder, 1/2 tsp baking soda, 1/2 tsp salt
Fold the dry ingredients into the wet, alternating with the milk in two additions, starting and ending with the dry. Mix until just combined. 80 g whole milk