Prepare coals or preheat grill. Place tomatillos, red onion and jalapeño directly on grill (without any oil). Salt them as they begin to char. Cook until charred so that black spots speckle the surface of the pepper, the tomatillos begin to burst and the onion slices begin to soften.
Remove from grill, allow to cool and reserve any liquid that accumulates. Once cool, roughly chop and add to food processor along with garlic, cilantro and lime juice. Pulse until smooth, using reserved liquid to thin as needed. Season to taste with salt. Set aside. (Can be made in advance and refrigerated up to 1 week.)
Preheat oven to 450°F. In a medium bowl, toss Pattypan squash and mushrooms with oil and season lightly with salt. Arrange in a single layer on a baking sheet and roast until squash browns slightly and releases some of its liquid and the mushrooms brown with edges slightly crisp, about 12—18 minutes.
Meanwhile, combine aioli ingredients in a small bowl, sifting paprika as you add it. Taste and adjust for seasoning.
To plate, toss warm vegetables lightly with salsa verde and taste for seasoning. Spoon some of the salsa verde onto the plate and top with a pinch of smoked paprika. Place vegetables on top of the salsa verde along with a few dollops of aioli and garnish with seasonal greens (purslane, wood sorrel or lambs quarters).
Recipe by Jonathan Dille, Executive Chef, The Grange, Providence