In a large bowl, whisk together the eggs, heavy cream, milk, sugar, vanilla bean paste, cinnamon, and salt until completely blended.
In a skillet, melt the butter and oil over medium low heat.
Soak a bread slice in the egg mixture for about 30 seconds to one minute per side.
Place the soaked bread in the heated skillet and cook until golden, about 2 minutes.
Flip and cook the other side for another 1-2 minutes, until golden. Try not to let the bread get too dark at this point. They will cook further with the sugar.
Set aside to a plate and repeat with remaining bread slices, adding more butter and oil as needed to keep a thin layer in the skillet.
Cover the top side of a cooked French Toast slice with about 1 tablespoon (12 grams) of granulated sugar in an even layer.
Add more butter and oil to the skillet if needed to create a thin layer and place the bread back to the pan, sugar side down.
Cook just until the sugar has melted and created a thin, caramelized layer. This should take 1-3 minutes. Lift the side occasionally to check the color and make sure the sugar doesn't burn.
Repeat with the remaining bread slices. Another option is to lightly torch the sugared top of each slice until caramelized or place the slices on a rimmed baking sheet under a broiler set to low.
Serve the Crème Brûlée French Toast warm with desired accompaniments such as fresh berries, whipped cream, maple syrup, and/or a dusting of powdered sugar.