- Place flour, sugar, salt, eggs, vanilla, zest, rum, orange blossom water, milk, and yeast in mixing bowl of stand mixer fitted with dough hook attachment. Mix for 5 minutes on medium speed until rough dough forms. Gradually add butter. Once combined, increase speed to medium, and continue mixing for about 10 minutes, until dough is smooth.
- Line sheet pan with parchment paper then sprinkle with flour. Place dough on pan and cover. Let rest for 20 to 30 minutes.
- Unwrap dough and knead by hand for another minute. Form smooth ball, cover, and let rise on floured work surface for 15 to 20 minutes.
- Roll out to ½-inch thick and then cut out doughnuts to desired size. Combine scraps and reroll to cut out more doughnuts. Let rise in warm place until doubled in size, about 1 hour.
- While doughnuts are rising, pour at least 2 inches
of oil in heavy pot and heat to 350 degrees over medium heat.
- Place doughnuts gently into pot and fry until golden brown, about 2 minutes on each side. Remove from oil and place on paper towels to drain. Toss in sugar and fill as desired with satsuma curd.
- Combine all ingredients together in large bowl. Place over double-boiler with simmering water and cook, whisking constantly, until mixture thickens and whisk leaves trail that holds shape.
- Remove from heat and strain. Cover with plastic wrap. Poke holes in wrap, and chill until ready to use. Can be made 1 day ahead.