In the bowl of a large food processor fitted with a dough blade or a large bowl, combine flour and salt. Slightly beat the egg, cube the butter, and add to the flour mixture. Slowly mix in the water using the food processor or your hands until dough comes together. Knead on a lightly floured surface until smooth and elastic. If too crumbly, add a little water. If sticky, add just enough flour to become workable.
Divide the dough into two pieces. On a lightly floured surface, roll the dough into a thin sheet (about 1/8 inch: 3 mm thick). Use a round cookie cutter or the top of a glass to cut out 2 3/4 inch wide circles.
Hull and cut the strawberries in halves or quarters for the large ones. Place a piece of strawberry in the center of each circle. Fold the circle in half over the strawberries and pinch together the edges to seal. Repeat until remaining dough is used.
In a large pot, bring water seasoned with salt to a boil. Place a few of the pierogi in the water. Don't overfill. Stir gently to keep them from sticking to the bottom. Once the pierogi rise, cook for an additional 2 minutes to heat the filling. Use a slotted spoon to transfer the pierogi to a serving dish.
Place dabs of softened butter over the pierogi, drizzle with heavy cream, and sprinkle with powdered sugar. Serve immediately.