My toddler recently developed a bit of intolerance to lactose. However, some of his favorite meals are moussaka and lasagna, not the least because the bubbly intensely flavored bechamel topping that roasts up in the oven. Typically for these recipes, I make a traditional white sauce or bechamel by melting butter and mixing in flour and cooking it out with very little color, then progressively whisk in milk until the sauce coats a spoon, then the final result is tempered with 3 eggs and shredded parm so that the baked product solidifies.
I can't find any suggestions on the internet for replacing the milk, butter, and cheese. Would olive oil in lieu of butter, and water or a nut milk (walnut or pine nut) instead of cow's milk, and omitting cheese suffice as a replacement? Is there any reason to believe the ratios would be out-of-balance using these substitutes?