In the bowl of a mixer fitted with a dough hook, combine 3 cups of the flour and the salt. Combine the yeast, sugar, and milk in a small bowl and let "proof" (foam up) for 5 minutes. Add the proofed yeast mixture to the flour. Add the eggs and butter and mix at low speed until the dough comes together and is soft, but not sticky. If the dough does seem sticky, add more flour, 1 tablespoon at a time, mixing after each addition, until a soft dough forms.

Turn out the dough onto a lightly floured work surface and knead for 1 minute, until smooth and elastic. Return the dough to the mixing bowl and cover with plastic wrap (or transfer to a clean oiled bowl and cover with plastic wrap). Let rise in a warm spot (such as an oven that is not on) until doubled in bulk, about 2 hours. (The recipe can be made up to this point and refrigerated after the kneading up to 1 day in advance. The dough will rise slowly in the refrigerator. Bring to room temperature before proceeding.)

Turn on the mixer for a few seconds to punch down the dough, or punch it down in the bowl with your fist. Turn out onto a lightly floured surface and knead a few times. Roll the dough out 1-inch thick, letting it "rest" a few times during the rolling (if you handle the dough too much, the natural gluten will tighten up and make your donuts tough).

Lightly flour 2 cookie sheets. Using a well-floured cutter, cut out hearts of dough, transferring them to the cookie sheets as you work, spacing them 2 inches apart. Reroll the scraps and cut out more disks to use up the dough. Cover lightly with a kitchen towel and let rise in a warm spot until doubled in bulk, about 20 minutes.

Meanwhile, pour 3 inches of vegetable oil into a deep, heavy pot and heat it to 350 degrees F. Lay out a thick layer of paper towels or brown paper bags for draining the fried donuts. Spoon the raspberry jam into your pastry bag. Working in batches of 6 or 7, transfer the risen donuts to the hot oil and fry until golden and puffed, turning frequently, about 5 to 7 minutes. Use a slotted spoon to transfer the hot donuts to the paper to drain. When cool enough to handle, poke the tip of your pastry bag into the center of each donut hole (entering from the side). Squeeze in a little jam. Sprinkle with powdered sugar and serve as soon as possible.