Preheat oven to 375 degrees F.
In a skillet heat 3 tablespoons of the butter. Add the hearts of palm, 1/2 teaspoon of salt and 1/4 teaspoon of the white pepper and saute until the hearts of palm are golden, about 3 minutes. Arrange the cooked hearts of palm in a layer on the bottom of a gratin dish.
In a saucepan melt 3 tablespoons of the butter. Add the shallots and garlic until softened, about 3 minutes. Stir in the flour, increase the heat to medium-high and cook for 2 minutes, being careful not to let it brown. Whisk in the milk and heavy cream, and cook, stirring constantly, until the mixture comes to a gentle boil. Cook until the floury taste is gone and sauce is smooth and thickened, about 3 minutes. Add the lemon juice, chives, remaining 1 1/2 teaspoons salt, and remaining 1-teaspoon of pepper. Add the lobster meat, and stir well.
In a small saucepan melt the remaining tablespoon of the butter and remove from heat. Add the breadcrumbs and toss to thoroughly combine. Pour the lobster mixture over the hearts of palm and spread the breadcrumbs evenly on top. Bake until golden brown and bubbly, about 20 minutes.
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