In medium saucepan over medium, heat oil. Add shallot and garlic; heat for 1 minute. Add strawberries, red wine vinegar, chipotle chili, and salt to pan and simmer over medium-low heat for 5 minutes. Stir in rosemary and remove from heat.
In skillet over medium-low heat, melt butter. Top a slice of bread with chicken, cheese, strawberry sauce, arugula, and a second slice of bread. Repeat with remaining ingredients. Heat until bread is crispy and golden on both sides and cheese has melted, about 3 minutes per side. Prepare sandwiches in batches if necessary.