I wonder if you could modify a recipe for cheese crisps.
The recipes I'm familiar with involve heating cheese through the melted stage and to a crispy crusty stage. It's usually done in disks or puddles, though the appearance is often airy or holed - from the bubbling of the cheese as it dries and sets, much like the lacy, caramelized cookies (Florentine, I believe).
And it seems from the recipes that it takes a bit for the cheese to crisp up after being taken off the heat, so you may be able to drape or fold the cheese directly from the heat so that as it cools it sets, crisps up, in rounded or arching configurations... and if you can't, you might be able to crisp them in such configurations using curved metal surfaces, like a metal or oven-safe glass bowl.
To move from the bubbly lacy disks to something more airy and cage-like, you would want long shreds - the cheese should be cut to matchsticks, rather than grated. Pre-shredded cheese is often dusted with starch which might help the shreds dry out by providing binding to the escaping oils... though it's probably not difficult to dust your own matchsticks if it does, in fact, help.
You'd have to scatter the shreds quite loosely to prevent them melting together - that is, with enough gaps to leave an impression of a net or cage. Because the space between means each strand or tangle is cooking more or less on their own on the baking sheet, you would need very short cooking times and a lot of vigilance to catch them while crisp and before burnt.