Dissolve yeast in a quarter cup of the water. Stir in one teaspoon of the sugar and let stand until foamy, 5-10 minutes.

Add the remaining water, remaining sugar, oil, eggs, and salt. Blend in one and a half cups of the flour. Add the remaining flour, a half cup at a time, until the mixture holds together. On a lightly floured surface, knead the dough until smooth and elastic, about 10 minutes.

Place in a greased bowl, turning to coat. Cover with a towel or plastic wrap and let rise at room temperature until doubled in bulk, about one and a half hours, or in the refrigerator overnight. (The dough is easier to handle when chilled, which is beneficial for cakes that require rolling.)

Punch down the dough. Divide the dough into thirds. On a lightly floured surface, roll out each dough piece a quarter-inch thick. Cut out three- to four-inch rounds.

Combine the poppy seeds, water or milk, sugar or honey, and salt in a small saucepan. Simmer over medium-low heat, stirring frequently, until the mixture thickens, about 10 minutes. Remove from the heat and stir in the remaining ingredients. Let cool. Store in the refrigerator if not using immediately.

Place one tablespoon filling in the center of each dough round. Pinch the bottom side of the dough round together over the filling. Fold down the top flap and pinch the other two sides together to form a triangle, leaving some filling exposed in the center. (The hamantaschen can be prepared ahead to this point and frozen for several months. Defrost before baking.)

Place the cookies two inches apart on several greased baking sheets. Cover and let rise at room temperature until nearly doubled, about one hour.

Preheat the oven to 350 degrees Fahrenheit.

Brush the tops of the hamantaschen with an egg wash (consisting of one egg beaten with one tablespoon water). Bake until golden brown, about 25 minutes. Transfer to a wire rack and let cool completely.