In a large bowl, whisk together the water, egg, vanilla, and melted butter until well combined.
Add 2 cups of flour (as a starting point), sugar, yeast, and salt, mixing until you have a very sticky mixture.
Add another cup of flour and stir until the mixture begins to come together into a rough dough. If needed, continue adding flour until the dough starts to form into a ball.
Knead the dough either directly in the bowl or on a lightly floured work surface, adding more flour as necessary to prevent sticking. Knead until the dough is smooth and elastic. When you poke the dough, it should spring back.
Shape the dough into a ball and return it to the bowl. Brush 1 tablespoon of oil over the top of the dough to prevent it from forming a skin.
Cover the bowl with plastic wrap or a lint-free tea towel, and set it aside in a warm place to rise for 45 to 60 minutes or until it has doubled in size.
Line a large baking sheet with parchment paper.
Once the dough has risen, punch it down to release the air. Transfer it to a lightly floured surface and divide it into 8 equal portions (or 12 for smaller donuts).
Roll each portion into a smooth ball and place them on the prepared baking sheet, ensuring they are at least 3 inches apart. If making 8 larger donuts, an extra-large baking sheet will fit them; otherwise, you may need two sheets for 12 donuts.
Cover the dough balls and let them rise again for 30 to 45 minutes, or until they have doubled in size.
Preheat your oven to 375°F.
Once the dough has risen, uncover it and use a small jar to press an indent into the center of each bun, going halfway down (be careful not to pierce through the bottom, or the filling will leak out). If you don’t have a small jar, you can use a similarly sized cup or even your thumb, but be sure to leave a little border to keep the filling in place.
Fill each indent with about 1 teaspoon of jam or until it is just below the top of the indent.
Bake the filled donuts in the center of your preheated oven for 15 to 20 minutes, or until the jam is bubbly and the edges are golden brown.
Allow the baked donuts to cool slightly to room temperature.
For the raspberry icing:
In a medium bowl, whisk together powdered sugar, jam, and 1 tablespoon of water until smooth and pourable. If the icing is too thick, add a little more water; if it's too thin, add more powdered sugar.
Drizzle the icing over the warm donuts and lightly dust the tops with powdered sugar if desired. Enjoy the donuts warm or at room temperature for the best experience.