Activate Yeast: In a small bowl, combine warm milk and yeast. Let it sit for 5–10 minutes until foamy.
Make Dough: In a stand mixer bowl, combine flour, sugar, and salt. Add eggs and yeast mixture. Mix using dough hook on low until combined.
Add Butter: Slowly add softened butter, one tablespoon at a time. Knead until dough is smooth, elastic, and pulls away from the sides (about 10–12 minutes).
First Rise: Transfer dough to a greased bowl, cover, and let rise for 1.5 to 2 hours, or until doubled.
Shape Donuts: Roll dough on a floured surface to ½ inch thickness. Cut out donut rounds. Place on parchment-lined tray. Cover and let rise again for 30–45 minutes.
Heat Oil: Heat oil in a heavy pot to 350°F (175°C).
Fry Donuts: Fry 2–3 donuts at a time, 1–2 minutes per side, until golden. Remove with slotted spoon to a paper towel-lined rack.
Make Salted Caramel: In a dry saucepan, melt sugar over medium heat. Stir continuously until amber. Add butter and mix until melted. Slowly pour in cream (it will bubble). Stir until smooth. Add sea salt.
Fill Donuts: Let donuts cool. Use a piping bag to inject caramel into the center or spoon over the top.
Finish & Serve: Dust with powdered sugar or top with more caramel and sea salt. Serve warm or at room temp.