Grease doughnut pan. This is easiest with some type of cooking spray.
In a large mixing bowl, whisk together flour, sugar, baking soda and salt, make a well in center of mixture and set aside. In a separate mixing bowl, whisk together milk, vegetable oil, lemon and eggs (mixture may appear to curdle a bit which is fine), then pour into well in flour.
Using a wooden spoon, stir mixture just until combined, then fold in 2/3 cup finely chopped strawberries.
Spoon batter into each well in the doughnut pan until it is about 2/3 of the way full. Bake in preheated oven 11 - 13 minutes until toothpick inserted into center of doughnut comes out clean.
Allow cooling for several minutes in doughnut pan then invert onto a wire cooling rack to cool completely. Once cool dip top halves of doughnuts into glaze and allow some of the excess to run off, then return to a wire rack, the glazed side facing upward. In a hurry? You can transfer doughnuts to the freezer to allow glaze to set, about 5 minutes.
In a bowl whisk together the sugar, strawberry puree, and milk. Add more milk if necessary to get to the desired consistency. If you find your glaze to be too runny you can add a little more sugar to thicken it up.
Use your favorite gluten free flour mix. I used Gluten Free Bisquick and it worked great.