Step 1

Melt butter in a large pot over medium-low heat. Add onion and 1/2 teaspoon kosher salt. Cook, stirring frequently, until onion is tender but not browned, about 10 minutes.

Step 2

Increase heat to medium-high. Stir in tomato paste. Cook, stirring frequently, until mixture begins to caramelize and brown, about 2 minutes. Add squash, chicken broth, remaining kosher salt, and cayenne pepper. Reduce heat to medium-low and simmer until squash is very tender, 20 to 25 minutes. Reduce heat to low.

Step 3

Using an immersion blender, blend squash mixture until smooth. (Or transfer mixture, in batches if needed, to a blender or food processor. Cover and blend or process until smooth. Return to pot.) Stir in cream and maple syrup.

Step 4

Ladle soup into wide shallow bowls. Serve with French Onion Toast Toppers and top with parsley.

Step 5

Preheat the oven to 425 degrees F (220 degrees C).

Step 6

Heat butter and olive oil in a very large skillet over medium-low heat until butter is melted, 1 to 2 minutes. Add sweet onions, shallot, salt, and pepper. Cook, covered, stirring occasionally, until onions are tender, 12 to 14 minutes. Increase heat to medium-high. Cook, uncovered, stirring frequently, until onions are golden brown, 3 to 4 minutes.

Step 7

Reduce heat to medium. Stir in sherry and cook 2 minutes more. Add low-sodium beef stock and fresh thyme sprigs. Bring to a boil. Reduce heat and simmer until liquid is reduced and onions are very tender, about 5 minutes. Remove thyme and discard.

Step 8

Meanwhile, arrange sourdough bread slices on a large baking sheet. Spread each slice with 1 teaspoon softened butter. Bake, turning once halfway through, until light brown, 8 to 10 minutes. Preheat the oven's broiler.

Step 9

Top bread slices with caramelized onion mixture and shredded Gruyère cheese. Broil until cheese is melted, about 2 minutes.