Preheat the oven to 425 degrees F and place a rack on the second-to-top position. Line a rimmed baking sheet with foil leaving a 2-inch overhang on each side. Set aside.
Heat 1 tablespoon canola oil in a medium saucepan over medium heat. Add the garlic, chiles and onions, sprinkle liberally with salt and cook, stirring occasionally, until starting to soften but not browned, 6 to 8 minutes. Add 1 cup water and the piquillo peppers and stir to combine. Bring to a boil, and then lower to a simmer and cook uncovered until the water has reduced by half and the vegetables are completely softened, 10 to 12 minutes.
Turn off the heat, stir in the vinegar and let cool for 5 minutes. Transfer to a blender and blend until completely smooth, 1 to 2 minutes. Season with more salt if needed. Reserve.
Heat the remaining oil in a large straight-sided skillet over medium-high heat until the pan is just starting to smoke. Sprinkle both sides of the chicken thighs liberally with salt. Brown the chicken thighs on both sides in 2 batches, 4 to 6 minutes total per batch, and then transfer to a plate.
Turn off the heat under the skillet and add the butter, swirling the pan until melted. Stir in the reserved hot sauce and return the chicken to the skillet, turning to coat. Bring the mixture to a boil, and then lower to a simmer and cook covered for 20 minutes. Uncover and continue cooking for 10 minutes longer.
Turn off the heat and transfer the chicken to a cutting board. Let cool for 5 minutes before cutting the thighs into bite-size pieces. Return the chicken to the hot sauce and stir to combine. Cover with a lid to keep warm.
Mix together the sour cream and blue cheese in a small bowl using a spoon until mostly smooth. Season with salt and black pepper. Reserve.
Spread the tortilla chips onto the prepared baking sheet in a single layer, allowing some overlap. Scatter the mozzarella cheese evenly on top, and then bake until the cheese is melted and the chips are warmed through, 2 to 3 minutes.
To serve, spoon the warm chicken thighs and hot sauce evenly all over the chips and dollop with half of the blue cheese sour cream and garnish with the sliced celery. Serve with the remaining blue cheese sour cream on the side.
Cook's Note: Make sure your skillet is really smoking before adding the chicken to avoid sticking.