Chocolate-filled Brioche Doughnuts – the glass monocle

Ingredients

• 2 quarts sunflower or canola oil ()
• 12 oz all-purpose flour ((about 2 cups))
• ¼ oz active dry yeast ((1 packet))
• 3 oz butter (with at least 82% fat) ((¾ stick))
• ½ lemon
• 1-½ tsp dark rum
• ¼ tsp nutmeg
• ¼ tsp vanilla extract
• 3 oz eggs ((1 or 2 eggs, weighed with shell))
• 1.5 oz standard, granulated sugar ((about 6 tbsp))
• 1 tsp more sugar ((for activating the dry yeast))
• ¾ tsp salt ()
• ½ cup water
• 2 oz standard, granulated sugar ((about ½ cup))
• 1-½ tsp ground cinnamon
• 8 oz confectioner's/powdered sugar ((about 2 cups))
• 4 oz butter (with at least 82% fat) ((1 stick))
• 2 oz unsweetened cocoa powder
• ½ tsp vanilla extract
• ½ tsp fine sea salt ()
• ¼ cup water ((2 fl oz; 4 tbsp))