An hour before cooking, pat the chicken dry with paper towels and season lightly with salt and pepper. Arrange the chicken skin-side up on a lightly greased sheet pan — place in the refrigerator, uncovered, to dry. Heat the oven to 450°F (230°C). Remove chicken from the fridge and drizzle with oil. Roast until the chicken is golden and has rendered its fat, about 30 minutes.
Take the sheet pan out of the oven. Dip the slices of bread in the rendered fat, turning to coat both sides. Leave the slices of bread in the pan and return to the oven. Then roast for an additional 10 minutes. The bread will be perfectly toasted.
While the chicken roasts, combine the brown sugar, colatura di alici, and lemon juice in a small saucepan and bring to a brisk simmer over medium-high heat. Cook until the mixture has reduced by about half, 3-4 minutes; the bubbles will increase in size — this is how you can tell it’s ready. Add both pieces of butter and stir them into the sauce until fully melted. Set aside.
To serve, place the pieces of toast on four (4) warmed serving plates. Divide slivered basil between the toasts, and top each one with a piece of chicken. Spoon the sauce over the chicken. The bread will absorb some of the liquid. Top each chicken thigh with fresh basil leaves and serve.