Make the chipotle sauce by combining the chipotle chile and 1 tsp adobo sauce, mayonnaise, Greek yogurt, chopped cilantro, ground cumin, and dried dill in a blender or food processor; puree until smooth, about 2–3 minutes. Refrigerate until serving.
Caramelize the onion: heat a small skillet over medium, add 1 tablespoon butter and the sliced onion, and cook gently, stirring occasionally, until soft and deep golden, about 15 minutes. Set aside.
Cook the chicken cutlet: heat a small heavy-duty pan over medium-high with 1 teaspoon oil. Season the cutlet with salt and pepper and cook 2–3 minutes per side until golden.
Add 1/4 cup water to the pan, reduce heat to low, and simmer the cutlet for 5 minutes. Remove to a bowl and let cool slightly, then shred the chicken with two forks until you have about 1 cup.
Assemble sandwiches: spread an even layer of Gruyère on two slices of bread. Top each with the shredded chicken and some caramelized onions, drizzle about 2 teaspoons chipotle sauce over the chicken, then add another layer of cheese and close with the remaining bread slices.
Heat a griddle or skillet over medium 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.