Warm the milk to 105–115°F (40–46°C) in a small saucepan or microwave; it should feel pleasantly warm, not hot.
Sprinkle the yeast over the warm milk and let stand until foamy, 5–10 minutes.
In a large bowl (or stand mixer bowl), whisk together the flour, granulated sugar, and salt. Add the eggs and vanilla, then pour in the foamy mixture. Mix with a spoon or dough hook until a shaggy dough forms, 1–2 minutes.
Knead until smooth and slightly tacky, 6–8 minutes by mixer or 8–10 minutes by hand. Add the butter a little at a time and continue kneading until fully absorbed and the dough is elastic and smooth, 3–5 minutes more.
Shape the dough into a ball, place it back in the bowl, cover, and let rise in a warm spot until doubled in size, 60–90 minutes.
Gently deflate the dough and turn it out onto a lightly floured surface. Divide into 16 equal pieces and roll each into a tight ball. Arrange on a parchment-lined baking sheet with space between, cover, and let rise until noticeably puffy, 30–45 minutes.
Pour the vegetable oil into a heavy pot to a depth of about 2 inches and heat to 350°F (175°C). Fry 4–5 doughnuts at a time, turning once, until deep golden and cooked through, 2–3 minutes per side; adjust the heat to maintain 340–350°F. Drain on a rack or paper towels.
When cool enough to handle, pierce the side of each doughnut with a skewer to make a channel, then pipe strawberry jam into the center until each feels slightly heavy, about 1–2 teaspoons per doughnut.
Dust generously with powdered sugar and serve warm the day they are made.