Chop finely the onions. Heat the oil in a pan. Add the onions and cook them on medium heat until translucent and soft (about 10 minutes), depending how much you like your onions cooked.
If adding the wild garlic, chop the wild garlic stems and leaves. Alternatively you can add minced garlic. Add the chopped wild garlic to the pan with the onions for the last minute of cooking the onions. Let onions cool (slightly)
Beat the eggs in a bowl. Add salt and pepper.
Moisten the matzo under running water until the matzo softens but does not crumble. Break the matzo in pieces and add them to the seasoned beaten eggs. Continue until all matzo is added. Add onion-garlic mixture to the beaten eggs. Leave for a few minutes until the matzo absorbs most of the beaten eggs.
Add the oil to the same pan over a medium flame. Add the matzo-egg mixture into the pan when it is pretty warm (like you would for eggs). If you are adding the grated cheese, add it now on top of the mixture. Cook 2-3 minutes before stirring and cooking for another 1-2 minutes until set.
Serve the matzo brei with toppings (see notes)