In the bowl of a stand mixer, attached with the dough hook, combine the warm water, a pinch of the sugar and yeast. Let stand for 5 minutes or until yeast has foamed up. Add 4 1/2 cups flour, warm milk, butter, eggs, sugar, and salt to the bowl and mix on low speed for 3 minutes. Add as much of the remaining flour as needed to create a smooth soft dough that pulls away from the sides of the bowl. Knead on moderate speed for 8 minutes. Transfer the dough to a greased bowl, cover loosely with plastic wrap and a damp kitchen towel. Set in a warm spot and let rest until doubled in size, about 1 hour.

Dump out the dough onto a generously floured large cutting board or surface. Roll the dough out to a 1/2-inch thick rectangle. Cut the doughnuts using a 3-inch round cookie cutter, and cut out the center with a round 1/2-inch cutter. Transfer the doughnuts and doughnut holes to a greased baking sheet and cover with plastic wrap and let proof at room temperature for another 30 minutes.

Meanwhile, fill a large heavy-duty pot halfway with vegetable oil and heat to 325°F. Fry the doughnuts in batches until golden brown on each side, about 5 minutes total. Transfer to a paper towel lined baking sheet or a wire rack, set over a baking sheet, to drain.

For the glaze, whisk together the powdered sugar, strawberry purée, heavy cream, and salt until smooth. If it's too thick add a splash or two of cream. If it's too thin add a bit more powdered sugar.

For the topping, combine the cookies and strawberries in a plastic bag. Using a rolling pin crush the cookies until coarse crumbs form. Transfer to a bowl and stir in the melted butter.

Dip each doughnut into the glaze and then into the topping to fully coat. Allow the glaze to set before serving. As with most fresh doughnuts, these are best eaten the day they are made. Enjoy!