Heat a large braiser or high-sided skillet over medium-high heat. Add 1 tablespoon olive oil and swirl to coat the skillet. Once the oil is shimmering, add the sausages and brown on all sides until browned but not cooked through, 5 to 7 minutes. Remove them to a plate and let cool until cool enough to handle, then slice them into 1 1/2-inch pieces. Set aside.
Turn the heat down to medium-low and add the remaining 2 tablespoons olive oil to the skillet. Add the peppers and onions and season with 1 teaspoon salt and 1/4 teaspoon pepper. Cook the peppers and onions, stirring occasionally, until they begin to soften, about 5 minutes. Add the garlic and stir to incorporate. Continue cooking the vegetables over low heat until they’re very soft, beginning to caramelize and have some spots of browning, about 40 minutes. If the vegetables start to look like they’re getting too browned too quickly, add a splash of water to deglaze.
Add the apple cider vinegar and the sausages to the skillet and stir to incorporate. Continue cooking until the sausages are cooked through, 10 to 15 minutes. Turn the heat off.
Garnish with the basil and serve immediately.
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