If you're using quark cheese, you can just mix everything in a bowl with a hand mixer. That works too with ricotta cheese and queso fresco, although the final texture will be a bit grainier.
If you're using cottage cheese, you'll need a blender and this works well with ricotta and queso fresco too.
Dump your cheese, heavy cream, honey, vanilla, starch, salt, and lemon zest into your mixing bowl or blender bowl.
Separate the eggs and add the yolks to the bowl with the other cheese batter ingredients. Place the egg whites into a very clean mixing bowl and add the salt
With a hand mixer and clean beaters, beat the egg whites on high till it's white and very thick. You'll see peaks forming and you'll be able to cut into it with a knife. This can take a few minutes.
Now beat or blend the cheese cake ingredients till smooth.
Add the fluffy egg whites to the cheese mix. If you were blending the cheese mix, pour it into a bowl before adding the stiff egg whites.
With a spatula, fold the egg whites into the cheese mix till well combined. If you have lumps, you can also gently break them up with a wire whip.
Pour the cheese mix over the prepared crust. It will be runny if you use cottage cheese.
Sprinkle the blackberries over the cheese mix. They'll sink right into your cottage cheese filling, but will lay on top if you've used any of the other cheeses. If you like, you can press the berries into the cheese topping just slightly.
Take your crumbles and break them up. Spread over your cheesecake.
Bake on the lower rack of your oven for about 45 minutes for a 9x13" pan. Check after about 30 minutes to see if it browns on top and cover it with some aluminum foil if it's getting darkish. It might take a little longer if you're using a 7x10" pan.Once the center is set up and you feel resistance when pressing on it, you can turn off the oven and leave it in there to cool down with the oven door slightly ajar. Cool completely before serving. I like to refrigerate mine for a few hours before I eat it.
This will stay delicious in the refrigerator for 4-5 days. And it can also be frozen and kept for 3 months in the freezer, wrapped tightly with plastic wrap and aluminum foil.