Preheat the broiler. Brush the bread with olive oil and transfer to a baking sheet. Broil the bread a few inches from the heat, turning once, for 2 minutes, until golden and toasted. Lightly rub the whole garlic clove over the toasts.

In a large, deep pot, heat the 1/4 cup of olive oil. Add the kielbasa, shallot, and sliced garlic, season lightly with salt and pepper and cook over high heat, stirring, until the garlic is softened and lightly browned, about 3 minutes. Add the mussels and cook, stirring, for 1 minute. Add the lager, cover and steam the mussels until they open, about 5 minutes. Remove from the heat. Using a slotted spoon, transfer the mussels to 4 deep bowls, discarding any mussels that do not open.

Add the butter, mustard, and dill to the broth, swirling and shaking the pot until the butter melts. Slowly pour the broth over the mussels, stopping before you reach the grit at the bottom of the pot. Serve the mussels with the garlic toasts.