In a large skillet or sauté pan, whisk together the vegetable broth, white wine, shallots, Dijon mustard, and capers.
Bring the mixture to a lively simmer over medium-high heat.
Gently place the whole hearts of palm into the simmering broth.
Season with a pinch of salt and pepper.
Reduce the heat to maintain a gentle simmer, cover the pan, and cook for 8-10 minutes, allowing the hearts of palm to heat through and absorb the flavors.
Using tongs, carefully transfer the hearts of palm to a plate and cover to keep warm. Increase the heat for the broth to medium-high and bring to a vigorous simmer for about 3 minutes to reduce and concentrate the sauce slightly.
Remove the pan from the heat.
Stir in the fresh lemon juice and chopped dill.
Taste and season with additional salt and pepper, if desired.
To serve, divide the hearts of palm among four shallow bowls. Ladle the warm broth around them and garnish with an extra sprig of fresh dill.