Make the pesto: Toast the almonds in a skillet over medium heat, tossing, until lightly golden, about 5 minutes. Transfer to a food processor.
Bring a medium saucepan of salted water to a boil. Fill a large bowl with ice water. Add the broccoli to the boiling water and cook until bright green, about 2 minutes. Drain and transfer to the ice water to cool, then drain and pat dry. Transfer the broccoli to the food processor with the almonds; add the basil, garlic and 1/2 teaspoon salt and pulse to make a chunky paste. With the motor running, gradually add the olive oil. Add the parmesan and pulse to combine. Transfer the pesto to a bowl and press plastic wrap directly onto the surface; set aside until ready to use.
Make the soup: Preheat the broiler. Put the bell peppers on a baking sheet and broil, turning, until charred, 7 to 10 minutes. Transfer to a bowl, cover tightly with plastic wrap and set aside 5 minutes. Peel the peppers with your fingers under running water; discard the stems and seeds. Roughly chop the peppers.
Heat the olive oil in a medium pot over medium heat. Add the celery, carrot, leek and 1/2 teaspoon salt and cook, stirring occasionally, until the vegetables are soft, about 7 minutes. Stir in the thyme and roasted red peppers. Push the vegetables to one side of the pot; add the tomato paste to the other side and cook, stirring, 2 minutes, then stir into the vegetables. Increase the heat to high; add the chicken broth and 2 cups water and bring to a simmer. Add the potato and simmer until soft, about 30 minutes. Puree with an immersion blender or in a regular blender in batches; keep warm.
Preheat the oven to 350 degrees F. Cut the bread into tree shapes using a cookie cutter. Transfer to a baking sheet and toast in the oven until crisp, about 6 minutes. Ladle the soup into bowls. Spread the pesto on the toast and float in the soup. Serve immediately.