Preheat broiler and place a rack in the upper third of oven.
Arrange tomatillos, onions, garlic and poblanos on a baking sheet, lightly dress with olive oil spray or EVOO and season with salt and pepper. Place under broiler and char until tomatillos have browned and burst, the onions are tender-crisp and charred at edges, and the garlic is golden. Let cool.
Peel and seed poblanos then coarsely chop. Remove garlic cloves from their jackets and the tops from peppers. Place everything in a food processor along with the thyme, oregano, cumin, lime juice, cilantro or parsley, and agave or honey. Pulse until almost smooth. Pour the mixture into a soup pot or Dutch oven; add stock and 1-2 cups water, depending on how thick or thin you prefer your soup. Bring soup to a boil then reduce heat to a low simmer.
Remove skin from chicken and pull meat into bite-sized pieces, discarding bones and skin. Add pulled chicken to soup.
Heat a large skillet over medium-high heat and coat lightly with spray or drizzle with oil. Add a large tortilla and char; flip, dot with crème fraiche and top with some of both cheeses and sliced pickled jalapeño peppers. Fold in half, press then turn; cook 1-2 minutes total to desired crispness.
Serve soup in shallow bowls with wedges of quesadilla alongside for dipping.
: If serving for brunch, add a medium-poached egg to soup.
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