Peel and cut squash into 1-inch (2.5-cm) cubes. Bring to a boil; reduce heat to medium and cook until soft, about 15-20 minutes.
In a medium saucepan, bring butter to a boil; heat until butter turns brown and smells nutty; remove from heat and strain through a fine mesh strainer. Add squash, salt and pepper and purée until smooth.
Spread 1 tbsp squash purée on inside of each slice of bread. Place half of cheese mixture on one side; place avocado on top of cheese; put remaining cheese on top.
Place second slice of bread on top; butter outsides of both slices. Grill on a panini grill for about 4 minutes.