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Step 1
In a medium saucepan, cover the potatoes with water. Set over high heat and bring to a boil. Cook at a boil, adjusting the heat as needed, until tender, 12 to 20 minutes. (The cooking time will vary according to the size of the potatoes.) Drain the potatoes and set aside.
Step 2
While the potatoes are cooking, in a 4-quart Dutch oven, soup pot or enameled cast-iron casserole over medium heat, heat 1 tablespoon of the oil until shimmering. Add the leeks, season to taste with salt and pepper, and cook, stirring occasionally and adjusting the heat as needed, until very tender but not browned, about 10 minutes.
Step 3
While the leeks are cooking, season the chicken lightly with salt and pepper. In a large nonstick skillet over medium heat, heat the remaining 1 tablespoon of oil until shimmering. Add as many of the tenderloins as will fit without crowding and cook until nicely browned on one side, 2 to 3 minutes. Turn over and cook until browned on the other side, about 2 minutes. Transfer to a plate and repeat with the remaining tenderloins. (The chicken does not need to be cooked through at this point.)
Step 4
When the leeks are soft, add the wine. Increase the heat to high, bring to a boil and cook until reduced by half, 3 to 4 minutes. Add the browned chicken tenderloins and broth, and bring to a boil. Reduce the heat to medium-low, cover and cook for 15 minutes, adjusting the heat so the liquid is very gently simmering. Taste, and season with more salt and/or pepper, if desired.
Step 5
To serve, divide the potatoes among shallow bowls. Top with the chicken and leeks, spoon the cooking liquids over the chicken and potatoes, and garnish with the chopped chives.