Set up the oven for rotisserie roasting and preheat the oven to 450 degrees F.

Smash the garlic cloves, sprinkle with 1 ½ teaspoons salt, and, with the flat side of a knife, mash and smear the mixture to a coarse paste. Melt the butter in a small saucepan. Stir half the garlic paste, the paprika, ground porcini, caraway seeds, and a generous amount of pepper into the melted butter and cook over medium heat, stirring until fragrant, about 30 seconds. Cool. Smear the chicken cavity with the remaining garlic paste, then stuff with the onion and lemon zest. Brush the spiced butter over the chicken and season with salt.

Tuck the chicken wings under the back, and cross the legs with kitchen twine. Put the chicken on the spit and secure tightly. Roast until golden brown and an instant read thermometer inserted in the thigh registers 170 degrees F, about 1 hour. Transfer chicken to carving board and let rest for 15 minutes before carving.

Wisk the horseradish into the sour cream in a small serving dish. Carve the chicken and serve with the sauce.