Heat 2 tablespoons of the olive oil in a medium saucepan over medium-high heat. Add the shallots and garlic and cook, stirring occasionally, until they begin to soften and caramelize, about 3 minutes. Add the remaining peppers, pepper trim, cumin, coriander, and piment d’Espelette, and cook, stirring, until tender, about 5 minutes. Add the honey, vinegar, and tomato puree and bring to a simmer. Transfer the mixture to a blender and puree, streaming in the remaining 4 tablespoons olive oil, until emulsified. Pass through a fine-meshed sieve and season with salt; reserve, chilled.