Prepare a large bowl of ice water. Bring a large pot of water to a boil. Salt the water, add the broccolini and cook until crisp-tender, 2 to 3 minutes. Drain; transfer to the ice water to cool. Drain.
In a large cast-iron skillet, melt 1 tbsp. butter over medium. Add the mushrooms and cook, stirring occasionally, until browned, about 8 minutes. Add the shallots; season with salt. Cook, stirring occasionally, until the shallots are browned and softened, about 7 minutes. Add the wine. Cook, stirring often, until almost dry, about 3 minutes. Transfer the mixture to a food processor; pulse to puree.
In a small saucepan, combine the tomatoes, chiles, and 2 cloves garlic; add enough water to cover. Bring to a simmer, salt the water, and cook until the tomatoes are tender, 2 to 3 minutes. Drain; transfer to a blender and puree. Season the salsa.
In a large skillet, melt 2 tbsp. butter over medium-high. Add the broccolini, the remaining 4 cloves garlic, the raisins and pine nuts. Cook, stirring occasionally, until the broccolini is browned in spots, about 4 minutes. Season; cover to keep warm.
In a large nonstick skillet, melt the remaining 2 tbsp. butter. Working in batches, fry the eggs until the whites are set and the yolks are still runny, 3 to 4 minutes. Season. Divide tortillas among 4 plates. Top each with about 3 tbsp. mushroom puree, the broccolini mixture, an egg, a spoonful of salsa and the chives.