Preheat the oven to 350°F (175°C). Line a 9x5 inch (23x13 cm) loaf pan with parchment paper, leaving an overhang on the long sides to lift the loaf out later.
Melt the 4 oz (113 g) salted butter in a light-colored saucepan over medium heat. Continue cooking, stirring constantly, as it foams and the milk solids at the bottom turn golden and smell nutty, about 5-7 minutes.
Immediately pour the butter and all the browned bits into a large mixing bowl and let it cool until just warm.
Melt the 0.5 oz (14 g) butter in a skillet over medium heat. Add the diced apples, brown sugar, cinnamon, and salt.
Cook for 3-5 minutes, stirring occasionally, until the apples are glossy and slightly softened. If a lot of liquid has pooled, remove the apples with a slotted spoon and simmer the liquid until thickened, then stir the apples back in.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Whisk the granulated sugar into the cooled brown butter. Add the eggs one at a time, whisking well after each, then whisk in the vanilla.
Mix in half the dry ingredients, then half the yogurt. Repeat with the remaining dry ingredients and yogurt, mixing just until combined. Do not overmix.
Bake for 55-70 minutes, until a toothpick inserted in the center comes out with moist crumbs but no wet batter. If the top browns too quickly, tent loosely with foil.
Cool in the pan for 15-20 minutes, then lift out using the parchment and transfer to a wire rack to cool completely.
Weighed, not scooped. Butter, sugars, flour, and yogurt are given by weight — a scooped cup of flour can run 20% heavy, which is the difference between a moist loaf and a dry one. The 1 tsp of flour in the cinnamon swirl (3 g) stays as a volume measure; it is below the resolution of most home scales.
Cool the brown butter. If it is still hot when the eggs go in, they will scramble. Just warm to the touch is right.
Cool the apples too. Warm apples will melt the batter and sink through the loaf.
Apple choice. Firm, tart varieties hold their shape best — Granny Smith, Honeycrisp, or Braeburn. Avoid Red Delicious, which turns mealy.
Dice small. About ¼ inch (6 mm) pieces distribute evenly and cook through in the skillet.
Doneness. This is a dense, moist loaf. Check at 55 minutes, but expect the full 70 if your apples were juicy.
Glaze last. Glazing a warm loaf makes it soak in and disappear. Wait for it to cool fully.