Place the russet and red potatoes and ½ teaspoon of the salt in a medium pot and cover with cool water. Bring to a simmer over high heat. Reduce the heat as needed to maintain a simmer and cook until the potatoes are tender, 10 to 12 minutes. Meanwhile, brown the butter.

Melt the butter in a 12-inch skillet (preferable something with a lightly colored interior) over medium-high heat. Bring the butter to a simmer and cook until it browns and smells nutty, about 7 minutes. Pour off ¼ cup of the browned butter into a small heatproof bowl and set aside.

Return the pan with the remaining browned butter to medium heat and add the cabbage and the whites of the green onions. The pan will be quite full, but the cabbage will wilt, making stirring easier. Season with the remaining ½ teaspoon salt and pepper. Cook until tender and lightly browned, about 8 minutes.

Drain the potatoes and return them to the cooking pot. Add the milk and mash with a potato masher. The potatoes should retain a few small lumps for texture.

Add the chicken broth to the cabbage pan and scrape vigorously to remove any browned bits from the bottom of the pan. Cook until the broth is reduced, about 3 minutes more.

Add the warm cabbage and greens of the green onions to the mashed potatoes and fold to combine. Gently stir in the reduced chicken broth. Transfer the colcannon to a large serving bowl and make a well in the middle of the potatoes. Pour the reserved ¼ cup brown butter into the well and serve immediately.