In a small saucepan, combine ½ cup heavy cream, ½ cup whole milk, and ¼ cup granulated sugar. Heat over medium, stirring occasionally, until the mixture comes to a simmer. Remove from heat and cool the mixture to below 110°F.
Stir 1 teaspoon active dry yeast into the cooled cream mixture. Let the yeast bloom for 10–15 minutes, until the mixture becomes puffy and foamy.
In the bowl of a stand mixer, add 2 cups all-purpose flour and ¾ teaspoons kosher salt and stir briefly to combine.
Pour the yeast-cream mixture into the flour mixture. Attach the dough hook and knead on speed 2 for about 5 minutes, until the dough is smooth and pulls away from the sides of the bowl.
Lightly oil a glass bowl and oil your hands. Scrape the dough out of the mixing bowl, shape it into a ball, and place it seam-side down in the oiled bowl. Turn the dough once so the top is lightly oiled.
Cover the bowl with plastic wrap or a clean kitchen towel and let the dough rise in a warm place until it is a little more than double in size, about 1 hour.
Line a work surface with a little flour. Turn the risen dough out and roll it to about ½-inch thickness. Use a 3-inch round cookie cutter to cut out donuts and a 1½-inch round cutter to cut out donut holes.
Gather scraps, reshape into a ball, and let the dough rest for 10–15 minutes before rolling again. Repeat cutting until you’ve used all the dough (or stop and keep the donut holes if you prefer).
Pour oil for frying into a heavy pot to a depth that allows the donuts to float (use a thermometer). Heat the oil to 360°F and maintain a frying temperature between 350–375°F throughout cooking.
Fry 3–4 donuts at a time (depending on pan size) without crowding. Fry until golden brown, flipping once halfway through cooking so both sides brown evenly. Fry donut holes the same way in batches.
Using a slotted spoon, transfer fried donuts and holes to a wire rack set over a baking sheet to drain excess oil, or place briefly on paper towels. Let them cool slightly until warm but not hot before glazing.
To make the vanilla glaze: in a small mixing bowl whisk together 1 cup + 6 tablespoons powdered sugar, ½ teaspoon vanilla extract, 2 tablespoons milk, and a pinch of salt until smooth and pourable. Add a drop more milk if needed to reach dipping consistency.
To make the chocolate glaze: in a small mixing bowl whisk together ½ cup powdered sugar, ¼ cup dutch-processed dark cocoa powder, a pinch of salt, ¼ teaspoon vanilla extract, and 7 teaspoons whole milk until smooth and pourable. Adjust by whisking until the glaze has a consistency suitable for dipping.
Dip each warm donut into the glaze of your choice, letting excess drip back into the bowl, then place glazed donuts on a wire rack set over a baking sheet to catch drips. Allow the glaze to set before serving.