Grease a 13" x 9" baking pan with butter. Arrange challah cut side down in prepared dish, making sure not to overlap; it’s okay if they fit a little snuggly.
In a large bowl, whisk egg yolks, egg, cream, vanilla, salt, and ⅓ c. sugar until pale yellow and smooth. Pour egg mixture over bread. Using your fingers, gently press so bread is submerged. Let soak 10 to 15 minutes.
Arrange a rack in center of oven; preheat to 300°. Cover dish with foil.
Bake French toast until custard mixture around bread is slightly firm and pale yellow, and an instant-read thermometer inserted into custard registers 170° (to ensure eggs are cooked), 40 to 45 minutes.
Sprinkle remaining ½ c. sugar on top. Using a blow torch, caramelize the top (or broil, watching closely, until caramelized, 8 to 10 minutes).
Let sit 10 to 15 minutes before serving.
