In a medium saucepan, combine the water, oats, cinnamon, and 1/4 teaspoon of the salt and bring to a boil over medium-high heat. Reduce the heat and simmer, uncovered, for 1 to 5 minutes (depending on the consistency you like it), stirring occasionally. Remove the pan from the heat and cover to keep warm.

In a large skillet, melt the butter over medium heat and cook, swirling the skillet occasionally, until the butter solids are just brown and have a toasty aroma, 3 to 5 minutes. Pour off all but 1 teaspoon of the browned butter into a small bowl and set aside.

To the hot skillet, add the pears, with cut sides down, and increase the heat to medium-high. Cook until nicely browned on both of the cut sides, turning once, about 5 minutes.

Decrease the heat to medium and add the cider, reserved brown butter, walnuts, honey, and remaining 1/4 teaspoon of salt. Simmer until the sauce reduces to a glaze that coats the pears and walnuts, 3 to 4 minutes.

Spoon the oatmeal into shallow bowls and top with the pears, walnuts, and sauce. Drizzle with milk or cream, if desired.