For the bread pudding: Butter a 13-by-9-inch baking dish with 1 tablespoon butter. Put the bread cubes in the dish.
Whisk together the half-and-half, 3/4 cup of the sugar, vanilla, 1 teaspoon of the cinnamon, salt and eggs in a large bowl. Pour this mixture over the bread. Cover and refrigerate for 1 hour.
Preheat the oven to 325 degrees F. Melt the remaining 3 tablespoons butter in a microwave or on the stove and drizzle over the top of the bread pudding. Combine the remaining 2 tablespoons sugar and 1/2 teaspoon cinnamon and sprinkle over the top.
Place on the center rack in the oven. Bake until the edges start to rise, a knife inserted into the center comes out mostly clean and the pudding jiggles when shaken, 45 to 50 minutes. Cool on a rack 35 to 40 minutes before serving.
For the sauce: Puree the berries in food processor. Press through a fine-mesh strainer. Stir in the honey and lime juice. Taste and adjust with more honey if it's not sweet enough for you.
Serve the bread pudding with a drizzle of sauce. Garnish with additional berries and a scoop of vanilla ice cream or whipped cream.