Preheat the oven to 425 degrees F. Mix the tomato juice, mezcal, lime juice, orange juice, adobo and oregano in a small pitcher; set aside. Arrange the jalapeno poppers on a baking sheet. Bake until golden brown, turning halfway through, about 15 minutes.
Meanwhile, place the tortilla chips in a resealable plastic bag and finely crush with a skillet or rolling pin. Mix with the chili powder, lime zest, orange zest and ? teaspoon salt in a small shallow bowl; set aside.
Melt 1/2 tablespoon butter in a small nonstick skillet over medium-high heat. Add 1 tortilla, then cover with the cheese and the remaining tortilla. Cook until golden on the bottom, about 2 minutes. Flip, adding the remaining 1/2 tablespoon butter to the skillet, and cook until the tortilla is golden and the cheese is melted, 2 to 3 more minutes. Transfer to a cutting board and cut into 6 wedges.
Thread the jalapeno poppers, quesadilla wedges and some pickled jalapeno slices onto the top half of two 12-inch wooden skewers. Moisten the rim of 2 pint glasses and press in the tortilla chip mixture. Fill the glasses with ice and the tomato juice mixture. Garnish with the skewers.