In a small saucepan, combine the milk with 4 of the cardamom pods, the vanilla bean and seeds, the cinnamon stick and 1/4 cup plus 2 tablespoons of sugar. Bring to a simmer, stirring until the sugar dissolves. Remove from the heat and let stand for 45 minutes. Strain the infused milk into a bowl and discard the solids.
Meanwhile, in a medium saucepan, combine the strawberries, framboise, lime zest and the remaining 1/4 cup of sugar and 2 cardamom pods and let stand for 10 minutes. Cook over high heat, stirring occasionally, until the berries are softened, 4 to 5 minutes. Let cool slightly. Discard the lime zest and cardamom pods.
In a large saucepan, melt the butter. Add the grits and cook over high heat, stirring until heated through, about 5 minutes. Gradually whisk in the warm water. Reduce the heat to moderately low and cook, stirring with a wooden spoon every 2 to 3 minutes, until the water has been absorbed and the grits are thickened, about 20 minutes. Whisk in the infused milk until smooth. Cook, stirring frequently with a wooden spoon until the grits are thickened, about 30 minutes longer. Pour the grits into a shallow 5- to 6-cup baking dish and let cool for 10 minutes.
Preheat the broiler and position a rack 8 inches from the heat. Sprinkle the brown sugar over the grits and lightly rub it into an even layer. Broil the grits until the sugar melts and caramelizes, about 5 minutes, shifting the dish as necessary for even browning. Alternatively, use a propane torch to caramelize the sugar. Spoon the grits into shallow bowls and top with a scoop of ice cream. Spoon the strawberry compote on the side.