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Preheat the oven to 400°F and have a large parchment-lined sheet pan nearby.
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For the breadsticks: In a small bowl, stir together the cream cheese, cheddar cheese, jalapeños, and garlic powder. Season with salt and pepper to taste.
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On a clean work surface, arrange 2 slices of bread. Spread the filling in a generous layer over the bread right to the edges and top with the remaining slices of bread. Generously butter both outer sides of the bread. On the sheet pan, sprinkle two piles of Parmesan in the shape of the bread. Carefully place the buttered breadsticks on each pile and slightly press them into place, then sprinkle more Parmesan over the top so that both sides are covered completely with cheese.
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Bake the breadsticks in the oven for 12 to 14 minutes, until the cheese is golden brown and crispy on both the tops and bottoms of the sandwiches and the inner cheese is gooey.
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Meanwhile, make the soup: In a high-powered blender, place the sun-dried tomatoes and crushed tomatoes and blend on high until you have a smooth and creamy puree. Taste and add the salt and red pepper flakes (if using), as desired.
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In a medium saucepan, bring the blended soup to a boil over high heat, then reduce the heat to low and cover to keep warm.
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Remove the breadsticks from the oven, allow to cool for a few minutes to harden, then cut into strips. Spoon the warm soup into bowls and serve alongside the breadsticks.