In the bowl of a stand mixer fitted with the dough hook, combine the warm milk, active dry yeast, and honey or brown sugar; let sit 5–10 minutes until foamy.
Add the room-temperature butter, eggs, 3 1/2 to 4 cups all-purpose flour, and kosher salt to the mixer; knead until a smooth ball forms, about 4 minutes.
Cover the bowl and let the dough rise at room temperature until doubled in size, about 1 hour.
Butter a bundt pan. In a shallow bowl, mix the brown sugar and cinnamon. Place the melted butter in a separate bowl. Sprinkle the bottom of the bundt pan with 1 tablespoon of the cinnamon sugar and 1/4 cup of the blueberries.
Punch down the dough, divide it in half, then cut into about 28 pieces total. Dip half the pieces in the melted butter, roll in the cinnamon-sugar, and place them in the prepared bundt pan in an even layer.
Spoon half of the blueberry jam over the first layer of coated dough pieces and sprinkle with 1/4 cup blueberries. Repeat with the remaining dough pieces, jam, and remaining blueberries. Drizzle any remaining melted butter and sprinkle remaining cinnamon sugar on top.
Cover the assembled pan and let the dough rest 15–20 minutes while you preheat the oven to 375°F (190°C).
Bake the monkey bread 40–50 minutes until golden brown; if it browns too quickly, tent with foil. Allow to cool 5 minutes in the pan, then run a knife around the edge and invert onto a plate.
Meanwhile, make the icing: brown the 4 tablespoons butter in a small saucepan over medium heat 2–3 minutes until lightly toasted, then whisk together the browned butter, cream cheese, powdered sugar, vanilla, and a pinch of salt until smooth.
Drizzle the icing over the warm monkey bread or serve on the side for dipping.