Preheat the oven to 350 degrees F. Brush an 8-inch square baking dish with oil.
Bake the waffles on a baking sheet until lightly golden and slightly crunchy, 10 to 12 minutes. Cut each waffle in half.
Fill a small pot with the sliced potatoes, a large pinch of salt and enough cold water to cover the potatoes by 1 inch. Bring to a boil. Once the water starts simmering, cook the potatoes until tender, about 2 minutes. Strain the potatoes, and transfer to a large bowl.
Heat the oil in a large skillet over medium-high heat. Add the peppers, scallion whites and garlic, and cook, stirring frequently, until the peppers are just tender and browned in some spots, 4 to 5 minutes. Transfer to the bowl with the potatoes, add 1 teaspoon salt and a few grinds of pepper and toss to combine. Transfer to the prepared baking dish, press into a flat layer and sprinkle half the Cheddar over the top.
Whisk together the milk, eggs, 1 teaspoon salt and a few grinds of black pepper in a medium bowl.
Layer the sausage, cut-side up, on top of the potato-pepper mixture. Shingle the toasted waffle halves, in two rows, on top. Pour the custard down one side of the casserole dish (it won't cover the waffles). Sprinkle the remaining Cheddar over the top, cover loosely with foil and bake until the Cheddar has melted, the custard is set, the potatoes are tender and the sausage is cooked through, about 40 minutes. Uncover, let rest for a few minutes and sprinkle the scallion greens over the top.
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