In a medium pot, add the potatoes and enough cold water to cover by 1 inch. Cover the pot and bring to a boil. Salt the water, lower the heat and simmer the potatoes, uncovered, until fork-tender, 12 to 15 minutes.
While the potatoes are working, in a large skillet, heat the EVOO, 2 turns of the pan, over medium-high heat. Season the chicken with salt and pepper and cook in the skillet, turning once, until browned, 2 minutes total. Add the shallots and cook for 3 minutes. Add the vinegar and cook until slightly reduced, 1 minute. Add the chicken broth, apricot preserves, honey and poppy seeds and simmer over medium-low heat, stirring. Stir in 1 tablespoon butter.
Drain the potatoes, return to their pot and add the remaining 2 tablespoons butter, the dill and salt and pepper to taste. Serve with the chicken.