Preheat the oven to 225 degrees F. Bake the chicken: Toss the chicken with the grapeseed oil and 1/4 teaspoon each salt and pepper in a medium bowl, coating well. Transfer to a baking sheet and bake until cooked through, about 35 minutes. Remove from the oven and let cool 1 hour.
Meanwhile, make the chutney: Combine the tomatoes, red onion, parsley, 1 teaspoon salt and 1/2 teaspoon pepper in a small saucepan and cook over low heat, stirring occasionally, 30 minutes. Remove from the heat and cover to keep warm.
Fry the chicken: Heat 3 inches of vegetable oil in a large pot over medium-high heat until a deep-fry thermometer registers 350 degrees F. Combine the flour, 1/2 teaspoon salt and a few grinds of pepper in a medium bowl, then add the chicken and toss to coat. Fry the chicken until golden brown, 4 to 5 minutes. Drain on a paper towel-lined plate. Let cool slightly, then cut each thigh in half.
Assemble the sliders: Spread a thin layer of butter on the cut sides of the rolls. Heat a large skillet over medium-high heat. Place the rolls, buttered-side down, in the pan and toast until golden, about 2 minutes. Spread with mayonnaise and sandwich with the chicken and chutney.