Prepare yeast mixture: Dissolve fresh yeast in lukewarm water with honey. Avoid hot water to prevent killing the yeast.
Mix dough ingredients: Add olive oil, egg, pinch of salt, baking powder, flour, and vanilla to the yeast mixture. Mix with a stand mixer paddle attachment or by hand until homogeneous, about 10-12 minutes to develop gluten.
First rise: Place dough in an oiled bowl, cover, and let rise at room temperature until doubled in size, about 1 hour depending on ambient temperature.
Shape donuts: Divide dough into 8 equal pieces. Shape each piece into a round donut, ensuring smooth surfaces for even rising and baking. Press the center to encourage hole formation.
Second rise: Let shaped donuts rise for 45 minutes to 1 hour until puffed up. Avoid pressing them down to keep the air inside.
Bake donuts: Preheat oven to 180°C (356°F) with the grill rack positioned high. Lightly brush donuts with olive oil. Bake for about 7 minutes until soft and puffed but not dried out.
Add toppings and fill: Brush warm donuts again with olive oil to help toppings stick. Optionally sprinkle with coconut flakes. Fill donuts with sugar-free strawberry jam using a piping bag or by cutting and spreading jam inside.