Whisk the eggs, yolks and sugar in a large bowl until smooth. Whisk in the cinnamon, nutmeg and salt until well blended. Add the milk, cream and vanilla; whisk until smooth. Add the croissant pieces, pressing to submerge them in the custard. Let sit for 15 minutes to allow the croissants to absorb the custard.
Preheat the oven to 350 degrees F. Grease the bottom and sides of a 6-cup jumbo muffin tin with the butter.
Stir the white chocolate chips into the croissant mixture, then evenly distribute the mixture among the muffin cups.
Bake until puffed and just set, about 30 minutes. Cool on a rack for about 10 minutes, then remove the bread puddings from the tins. Serve warm, with whipped cream if desired.