Pulse garlic, oregano, salt, and pepper in a mini food processor until finely chopped. Add oil and lemon juice; process until smooth.
Place chicken and garlic mixture in a large ziplock plastic bag. Seal bag, and let stand 30 minutes at room temperature or up to 8 hours in refrigerator. Remove chicken from marinade, letting excess drip off. Discard marinade.
Melt butter in a large, heavy-bottomed saucepan or Dutch oven over medium-high. Add chicken, breast side down, and sear until just browned, 3 to 5 minutes. Flip chicken, and sear until browned, 3 to 5 minutes.
Using tongs, transfer chicken to a plate. Reduce heat to medium, and add chorizo, brown sugar, and vinegar to pan. Cook, stirring often, until brown sugar is melted, 3 to 5 minutes.
Arrange chorizo mixture around edges of pan, and return chicken to pan, breast side down. Cover, reduce heat to low, and cook until evenly browned, about 30 minutes. Flip chicken, cover, and cook until a meat thermometer registers 185°F when inserted in thickest portion of thigh and 165°F in thickest portion of breast, about 30 minutes.
Transfer chicken to a cutting board; let rest 5 minutes. Pour any accumulated juices from cutting board into remaining chorizo mixture in pan. Remove chorizo, and thinly slice. Pour sauce mixture through a fine wire-mesh strainer into a serving bowl; discard solids. Carve chicken, and serve with sauce and sliced chorizo.
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