Place sweet potato on a foil-lined baking sheet. Bake for about 45 minutes or until softened. Cool to room temperature and puree in food processor.
Spray a muffin tin with cooking spray or oil. Divide bread pieces evenly among all 12 muffin cups.
Beat 3 whole eggs in a medium bowl. Add milk, ½ cup half-and-half, 1 teaspoon vanilla, and brown sugar and whisk until well-combined. Whisk in ½ cup of pureed sweet potato (use any remaining potato puree for whatever you like or just eat it as-is--whatever!).
Pour egg mixture over bread. Cover muffin tins with plastic wrap and refrigerate overnight.
To make the creme anglaise, heat half-and-half in a large heavy-bottomed saucepan over medium-low heat. Cook until just simmering, about 5 minutes.
In a medium bowl, whisk together sugar and egg yolks. Slowly add half of half-and-half mixture to the egg yolk mixture, whisking constantly. Pour half-and-half and egg mixture back into saucepan and raise heat to medium. Stir constantly with a wooden spoon until sauce has thickened and coats back of spoon, about 5 minutes. Remove from heat immediately and stir in vanilla extract and cardamom; pour through strainer into a heat-proof bowl. Cover and refrigerate overnight.
In the morning, preheat oven to 325°F. Remove plastic wrap from muffin tins and use a spoon to push bread down into the egg mixture before placing in oven. Bake for 35 minutes or until edges are golden but not brown. Loosen French toast cups by running a butterknife along the edges before popping them out. Serve with chilled creme anglaise.
Prep time does not include time spent overnight in fridge.