Preheat the oven to 400 degrees F. Whisk the eggs, heavy cream, 1/2 teaspoon salt and a few grinds of pepper in a large bowl. Heat 1 tablespoon olive oil in a large nonstick skillet over high heat. Add the poblano, season with 1/4 teaspoon salt and cook until charred in spots, 5 to 8 minutes. Add to the egg mixture along with the scallions and all but 2 tablespoons of the cheese; stir to combine.
Pour the enchilada sauce into the skillet and heat until bubbling; remove from the heat. Using tongs, dip the whole tortilla in the sauce until coated. Place in a 9-inch pie plate. Dip the tortilla halves in the sauce and arrange around the sides of the pie plate, with the rounded edge up. Reserve the remaining sauce.
Pour the egg mixture into the tortilla crust, sprinkle evenly with the reserved 2 tablespoons cheese and drizzle with 3 tablespoons of the remaining enchilada sauce. Bake until golden, slightly puffed and set, about 25 minutes.
Meanwhile, peel the jicama and cut into matchsticks. Toss with the lime juice, cilantro, remaining 2 tablespoons olive oil and 1/4 cup water in a bowl; season with salt and pepper. Reheat the remaining enchilada sauce over medium-low heat, thinning with a couple tablespoons of water if necessary. Cut the quiche into wedges and drizzle with the sauce. Serve with the jicama.