In a large bowl, whisk together the flour, instant yeast, 2 tablespoons of sugar, and salt. Add the warm milk, 1 tablespoon of melted butter, beaten egg, and vanilla extract. Stir until a shaggy dough forms and no dry flour remains, scraping down the sides of the bowl.

Cover the bowl with plastic wrap or a clean kitchen towel and let the dough rest at room temperature for 15 to 20 minutes. This rest helps the gluten develop naturally, making the dough less sticky and easier to handle.

Turn the dough out onto a lightly floured surface. Knead for 5 minutes, adding a little more flour if the dough is too sticky, but only as needed. Cover the dough with an inverted large bowl and let it rest for 5 minutes.

Knead the dough for another 5 minutes until it becomes smooth and supple. The dough should be soft and hold its shape; when you poke it firmly, the indentation should spring back quickly. If not, knead for a few more minutes.

Place the kneaded dough back in the bowl, cover with plastic wrap, and let it rise for 1 hour, or until doubled in size.

Turn the risen dough onto a floured surface. Roll it out into an 8 by 10-inch rectangle, about 1/2-inch thick. Use a 2.5-inch round cookie cutter or the rim of a glass to cut out 12 rounds. Gather and re-roll scraps if needed.

Preheat your oven to 375°F. Line a baking sheet with parchment paper or a silicone mat. Carefully transfer the dough rounds to the sheet using a scraper to keep their round shape. Lightly dust the donuts with flour, cover with a dish towel, and let them rest for 30 minutes at room temperature until nearly doubled in size.

Bake the donuts for 10 to 12 minutes, or until they are lightly golden. Check them near the end to prevent over-browning.

Place the 1/4 cup of sugar in a shallow dish. As soon as the donuts come out of the oven, brush them with the remaining 1 tablespoon of melted butter. Roll each warm donut in the sugar to coat evenly.

Fill a pastry bag fitted with a round tip with the strawberry jam. Use a knife to cut a deep slit into the side of each donut. Insert the tip and gently pipe about 2 teaspoons of jam into each donut. Serve warm or at room temperature.