What kind of cheese are you using? Some types of cheddar are pretty greasy when melted, for instance. I personally have no use for any burger that is listed as very lean; I hate the way really dry hamburger cooks, and just like in steak, fat adds flavor. Having said that, however, I almost never bake hamburger from the raw state, and therefore usually have a chance to pour off the fat before I use it in a dish.
Does your recipe start with raw hamburger, or browned? I'm assuming browned. That's the way I'd do it if I had those ingredients and was concocting this recipe, because otherwise everything else would be overcooked.
I therefore suspect your cheese is the culprit for greasiness. As far as helping with the greasy taste, I don't think there is anything that will really do that, because most of the time greasiness is not just a taste issue, it also has to do with mouth feel. A hot Salsa Mexicana (or Pico de Gallo, depending on what region of Mexico you're talking about, they can both be names for the same thing) would certainly cut the greasy mouth feel, but wouldn't prevent it in the first place.
If the cheese is not the culprit, one thing I'd be tempted to try is to buy some corn tortillas, nuke them to make them pliable, cut them to fit the mini-muffin tin, and line the forms with them before placing the burger in there. Any "grease" would help them to crisp, providing the missing crunch. In fact, I would be tempted to brush the cups in the tin with vegetable oil before putting the tortilllas in to aid in this process.
Just about any recipe can do with some tweaking to adjust it to your own tastes, unless you're baking. And while this recipe may seem like baking, it really seems more to me like assembling and reheating. Don't get me wrong; some of my favorite recipes work like this. My Tex-Mex style enchiladas, for example. Cook the filling meat, make the sauce, assemble, and heat everything in the oven for about a half-hour. Assembling, and reheating.
Good luck, and please let me know if any of this helped.