In a large bowl, add the white and brown sugars, cinnamon, nutmeg, and stir to combine.
In a saucepan, melt the vegan butter. Pour in the measured plant-based milk, coconut milk, and the bottle of JUST Egg. Heat over medium heat until warm but not boiling. This step prevents the melted butter from re-solidifying when added to the rest of the ingredients.
Add the vanilla extract to the warm liquid ingredients, then add the liquid mixture to the dry ingredients and whisk until everything is well-combined. Taste the mixture and adjust sugar/spice amounts to your liking.
Make the nutty topping: in a small saucepan, melt ¼ C vegan butter and stir in the chopped nuts, brown sugar, cinnamon, and salt. Stir it all together and set aside for assembly.
Preheat your oven to 375° F while you assemble the bread pudding.
In a large casserole dish, arrange the bread pieces and dried fruit (if using) in an even layer. Pour the liquid mixture over the bread and press bread pieces down so that all are hydrated by the liquid. (It's ok if some are popping out of the liquid - these will become crispy and toasted during baking).
Sprinkle/spoon the nutty topping evenly over the bread pudding. Bake for 50 minutes, turning your casserole dish halfway through baking. If the pecan pieces start to get too browned, cover with foil.
Allow the bread pudding to cool for 15 minutes before slicing. Serve warm or cold with coconut whipped cream or ice cream.