Bring a pot of water to a boil for the eggs and a separate pot of salted water to a boil for the potatoes.
Melt the butter in a medium saucepan over medium heat. Whisk in the flour and cook, whisking constantly, for about 2 minutes, until pale gold and lightly nutty-smelling.
Gradually whisk in the chicken broth, adding it slowly to avoid lumps. Add the bay leaf and allspice berries. Bring to a gentle simmer and cook for 8–10 minutes, until slightly thickened.
Remove the bay leaf and allspice berries. Stir in the vinegar, sugar, salt, and pepper. Taste and adjust the tang-to-sweet balance before continuing.
Reduce the heat to low. Stir in the chopped dill, then whisk in the sour cream a spoonful at a time, off direct heat or at the barest simmer, until smooth and glossy.
Meanwhile, boil the eggs for 10 minutes, then transfer to an ice bath and peel. Boil the potatoes in the salted water for 12–15 minutes, until fork-tender, then drain.
Halve the eggs and arrange with the potatoes in shallow bowls. Ladle the warm vomacka sauce generously over the top. Garnish with extra fresh dill and serve immediately.