Set up a food processor next to the stove.

Place the eggs and egg white in a glass measuring cup and beat with a fork to combine.

Place the water, butter, milk, sugar, and salt in a small saucepan set over medium heat and bring mixture to a boil, stirring occasionally with a wooden spoon. Remove from the heat and stir in the flour to combine; continue stirring with the wooden spoon until mixture comes together and clears the sides of the pan. Set pan over low heat and cook mixture, stirring constantly, until mixture is shiny and beads of fat appear in the bottom of the pan. (This will take about 3 minutes.)

Quickly transfer the mixture to the food processor and process for 10 seconds to cool a bit. With processor running, slowly pour the beaten eggs through the feed tube in a steady stream. Stop processor, scrape down the sides and bottom of the bowl, then process for 30 seconds more.

Preheat the oven to 425 degrees and adjust a rack to the middle position. Line a large baking sheet with parchment paper and set pan aside.

Fit a pastry bag with a 1/2-inch round tip and fill the bag with the pastry. Pipe into about 9 (2 1/2-inch) mounds on the parchment, spacing about 1 1/2 inches apart. (You can eyeball this or draw circles on the underside of the parchment to use as a guide.) Alternatively, you can just spoon the pastry onto the parchment.

Use your finger or a teaspoon dipped in water to smooth the tops of the mounds.

Bake for 17 minutes at 425, then turn oven down to 375 and bake 6 to 8 minutes more, until puffs are golden. (Don’t open the oven door during this process.)

Remove puffs from the oven and turn the oven off. Use a small, sharp knife to cut a 1-inch slit into the side of each of the puffs, then return them to the oven, propping the oven door open slightly. Let the puffs dry out in the cooling oven for 45 minutes. Transfer to a wire rack to cool completely. (At this point you can freeze the puffs in a zip-top bag for up to 1 month; heat them in a 300-degree oven for 10 minutes before serving.)

Process the 1/2 cup of strawberries in the bowl of a food processor until finely pureed; you should have about 1/3 cup of puree.

Beat the heavy cream in the bowl of a stand mixer fitted with the whisk attachment until it thickens. Add the sugar, vanilla, and food coloring (if using), then continue beating until it holds stiff peaks. Gently fold in the berry puree with a rubber spatula.

Split the puffs in half and fill each bottom half with filling and top with sliced berries (if desired). Top with the other half of the puff. Serve immediately.