Make the chipotle sauce by combining the chipotle chile, 1 teaspoon adobo sauce, mayonnaise, Greek yogurt, cilantro, ground cumin, dried dill, and kosher salt in a blender or food processor; puree until smooth, about 2–3 minutes. Refrigerate until ready to use.
Caramelize the onion: heat a small skillet over medium heat and add 1 tablespoon butter and the sliced onion; cook gently, stirring occasionally, until soft and golden brown, about 15 minutes. Remove from heat and set aside.
Cook the chicken cutlet: heat a heavy-duty skillet over medium-high and add 1 teaspoon oil. Season the cutlet with salt and pepper and cook 2–3 minutes per side until golden.
After browning both sides of the cutlet, add 1/4 cup water, reduce heat to low, and simmer for 5 minutes. Remove the cutlet and let it cool slightly, then shred it with two forks or your fingers into about 1 cup cooked shredded chicken.
Assemble the sandwiches: spread an even layer of Gruyère on two slices of bread. Top each with the shredded chicken and caramelized onions, drizzle about 2 teaspoons of chipotle sauce on each, then add more cheese and top with the remaining slices of bread.
Grill the sandwiches: heat a griddle or skillet over medium heat and add the remaining 1 tablespoon butter. Place a sandwich on the griddle and cook 2–3 minutes per side, pressing gently, until the bread is golden and the cheese is melted. Repeat with the second sandwich and serve warm.