Wrap the garlic, bay leaves, thyme, and half of the ginger together in a cheesecloth pouch.
Place a large pot over medium heat with the sugar. Cook the sugar, gently moving the pot around over the heat, until it reaches a caramel color. Add the carrots and onions and cook for 6 minutes over medium-low heat, stirring to prevent burning. Add the pineapple and stir for 1 minute. Add the tomato paste, salt, pineapple juice, tamari, and honey, stirring well. Add the pouch of herbs and water and bring to a boil. Cover and simmer for 40 minutes. Turn off the heat, remove the herb pouch and purée with an immersion blender until smooth.
Return the sauce to the heat. In a small bowl, combine the cornstarch with 1 tablespoon of the hot sauce to form a paste. Whisk the cornstarch slurry into the sauce and cook until thickened. Taste and adjust the seasoning. Cool the sauce fully.
When the pineapple sauce is cooled, coat the salmon in half of the sauce and marinate for 4-12 hours. Reserve the remaining sauce for the peppers.
Preheat the oven to 375°F. Place a rack in the top third of the oven.
Remove the salmon from the marinade to a baking sheet and place the remaining ginger slices on each piece of salmon. Roast on the upper rack of the oven until cooked through, about 15 minutes. For the last 1 minute of cooking, turn on the broiler to caramelize the top.
Bring the rice, 2 cups of water, and ½ teaspoon salt to a simmer in a small saucepan. Reduce heat to low, cover, and cook until the grains are tender and the water is absorbed, about 20 minutes.
Heat the oil for the peppers in a large skillet. Add the peppers and mushrooms and cook, stirring, until softened, about 8 minutes. Add the reserved pineapple sauce and toss to coat.
To serve, divide the jasmine rice between plates. Top with a salmon fillet and the peppers and mushrooms. Garnish with scallions.