Greg's enchiladas verdes o rojas

These are the recipes I'm gathering. The only real difference I see between enchiladas verdes and enchiladas rojas is the sauce, so I'm lumping them together here. I've only seen mention of the onion rings for enchiladas rojas, so maybe that's a difference.

Ingredients

For 12 enchiladas (6 portions)

quantity	ingredient	step
250 g	chicken breast	1
50 g	carrot	1
20 g	garlic, crushed	1
5 g	epazote	1
50 g	onion	1
1 g	pepper	1
	water	1

12	tortillas	2
40 g	lard	2
100 g	soft cottage cheese (optional)	2

1 litre	salsa verde or salsa roja	3
50 g	onion, cut into quarter rings (optional)	3
150 g	crumbly cheese (see discussion)	3

Preparation

Chop the carrots and onion coarsely. Crush the garlic and pepper. Put all ingredients in a saucepan, cover with ample water (the chicken will rise) and boil for 30 minutes. Leave to cool, then shred the meat. Keep the broth for the salsa verde or salsa roja.
Mexican recipes state that you should fry the tortillas individually in lard until soft. I find that by keeping them between two plates after cooking, they become soft enough without the lard. But then there's the flavour, so I melt lard and wipe the tortillas in it. Next place some of the chicken (and cheese, if used) on the tortilla and roll it up. Place in an oven-proof dish.
Put the onion slices over the enchiladas if using them, then pour over the salsa and bake in the oven for 30 minutes at 180° with grill on. The cheese can go on at the beginning or in the middle (and be melted) or at the end (and remain fresh), depending on the cheese.

Discussion

A number of things make this dish a problem, and I'm still not happy with it:

I'm still approximating to the quantities. Hopefully the ones here will be close.
I'm not sure what kind of cheese to use for step 3. The best Mexican recipe I have seen for enchiladas verdes asks for Queso Cotija, and I've seen suggestions that this might be similar to Feta. But Feta doesn't melt under the grill. I tried Haloumi, but it doesn't either. Maybe the soft yellow cheese that passes for Mozarella in Australia would be the best choice.
The tortillas tend to disintegrate. Somehow they need to be more solid, maybe by grilling before putting into the salsa.

Alternatives

There are a number of alternative approaches here, some of which I shall try. One Mexican recipe for enchiladas verdes, now succumbed to web rot, bakes the tortillas alone for 5 to 7 minutes, then dips them in salsa, folds them them on a base of salsa (“4 tablespoons or more”), adds the chicken and more salsa, then garnishes with cheese, lettuce, cream and cilantro. Baking might be a good idea: I've had perfectly normal tortillas almost disintegrate after being baked with salsa.

I've also taken some ideas from this recipe for enchiladas rojas.