Preheat oven to 350°F (175°C). Melt the 2 tablespoons salted butter; use about 1 teaspoon of it to grease a 6-cup doughnut pan (or a 12-cup muffin pan) and set the remaining melted butter aside for the batter.

In a large bowl whisk together the reserved melted butter, 1/3 cup whole milk (room temperature), 1/4 cup plain Greek yogurt, 1 large egg (room temperature), 2 teaspoons vanilla extract, and 1/4 cup granulated sugar until smooth and combined.

In a separate bowl whisk together 1 1/4 cups all-purpose flour, 1 teaspoon baking powder, 1/2 teaspoon baking soda, and 1/2 teaspoon kosher salt. Add the dry ingredients to the wet ingredients and stir until just combined (do not overmix).

Fold 2 tablespoons fresh or frozen blueberries into the batter.

Measure 1/4–1/3 cup blueberry jam (use 1/4 cup for lighter streaks or 1/3 cup for more color). Dollop tablespoon-size amounts of the jam into the batter and fold only 2–3 times so you keep streaks of jam rather than fully incorporating it.

Spoon the batter into a zip-top bag, snip a 3/4-inch corner, and pipe the batter into the prepared doughnut pan, filling each cup 1/2 to 2/3 full. (If using a muffin pan, fill each cup about 2/3 full.)

Bake in the preheated oven: 10 minutes for a doughnut pan, or 10–12 minutes for a muffin pan. Bake until lightly golden and a toothpick inserted into the cake portion comes out clean or with a few moist crumbs.

Remove the pan from the oven and let sit 5 minutes. Run a thin knife around the edges of each cup to loosen any berries that may have stuck, then invert the doughnuts onto a cooling rack. If you have more batter than fits in your pan, repeat filling and baking with the remaining batter.

While the doughnuts cool until warm (not hot), combine 1/2 cup granulated sugar and 3/4–1 teaspoon ground cinnamon in a shallow bowl. Melt the 4 tablespoons salted butter and keep it warm for brushing.

Working one doughnut at a time, brush the top and sides with the melted 4 tablespoons salted butter, then dip or toss the doughnut in the cinnamon-sugar mixture to coat. Place coated doughnuts back on the cooling rack and repeat with the remaining doughnuts.

Let the coated doughnuts set for a minute, then serve.