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In the bowl of an electric mixer fitted with a dough hook, mix together the flour, yeast, sugar, and salt on low speed until blended. Slowly drizzle in the butter and allow it to continue mixing for 3 minutes after you've poured in the butter. Then add the egg, vanilla, and buttermilk and increase the speed to medium. Allow the dough to knead for 8 minutes. Turn off the mixer, remove the bowl, cover it with a towel, and place it in a warm and dry location to rise for 1 and 1/2 hours.
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While the dough is rising, you can begin making the glaze. Preheat the oven to 375 degrees and place the rhubarb and strawberries in a small greased oven-safe casserole dish or pan. Roast them in the oven for 20 minutes, then remove and set them aside to cool enough for handling. While they are cooling, you can make the brown butter. When browning the butter, it is best to use a stainless steel one so that you can see the color of the butter change. Yes, I was naughty and used a black one but if you're doing this for the first time I highly recommend using a stainless steel one. Heat the butter in a small shallow frying pan over medium heat. Swirl the pan around a bit every couple minutes to help it cook evenly. Over a period of a few minutes, you'll notice the foam at the top of the butter start to change from light yellow to a dark tan. Once it reaches the dark tan stage and the butter looks light brown and golden, smell it. It should smell nutty and similar to toffee. Remove it from the heat and allow it to cool slightly.
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Pour the brown butter into a blender or food processor, leaving the hard burnt bits at the bottom of the pan. Add the strawberries, rhubarb, and cream and blend until a smooth puree is formed. Pour the puree into a small thick-bottomed saucepan and cook over low heat, whisking constantly until the mixture is very warm/barely hot. Remove the pan from heat, add the white chocolate chips, and whisk until the white chocolate has melted and is completely incorporated into the mixture. Place the mixture in the refrigerator to chill.
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While it is cooling, lightly grease the donut pan. Punch down the dough and divide it into 12 evenly-sized balls. Lightly flour them and then take a ball, poke a hole in the middle of it, and gradually shape it into a doughnut. Place it in the pan and repeat until you've ran out of dough or filled up the pan. Now let the dough rest for 30 minutes. If you only have one doughnut baking sheet that fits 6 doughnuts like myself, you don't have to let the dough rise for another 30 minutes when you put the second batch in the pan, you can just pop that tray in the oven straightaway.
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Preheat the oven to 350 degrees. After the 30 minutes of resting has passed, put the sheet in the oven and bake for 10-12 minutes or until the doughnuts have puffed up and are firm when you poke them but still white. You don't want them to get golden brown all over because at that point they've usually puffed up too much and closed up on the signature doughnut hole. If you find that your doughnuts have puffed up too much and closed in on the hole, do not fret! You can simply cut a hole in them with a knife once they've cooled. When you dip them in the glaze you won't be able to tell the difference.
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So backtracking a bit, once you remove them from the oven flip the pan over onto a wire rack and allow the doughnuts to cool completely. If you need to bake another batch of the doughnuts do so now. Once the doughnuts are completely cooled (this is important), then you can take the cooled glaze mixture out of the refrigerator.
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Whisk the powdered sugar into the glaze until it is completely combined. When it comes to glazing the doughnuts, there are two options. You can drizzle the glaze over them with a spoon, or just grab them between two of your fingers and dip the top half in so it has a nice even glaze coating. Sprinkle with poppy seeds as a garnish and serve! If you have leftovers, they keep well in the fridge for a few days when wrapped well with plastic wrap.