Mash minced garlic to form paste; place in bowl. Add vinegar, mustard, paprika, thyme and pepper; stir to combine. Set aside.
Starting at neck end of each chicken and working with hands between skin and breast, gently lift skin from breast, being careful not to tear. Lift skin away from thighs and legs. Using fingers, spread half of the garlic mixture over breast beneath skin of each chicken, working forward to legs. Place on rack in large roasting pan.
Roast in 400°F (200°C) oven for 45 minutes.
Pour vinegar over chickens. Roast until juices run clear when thigh is pierced and meat thermometer inserted in thigh registers 185°F (85°C), about 45 minutes. Transfer to cutting board and tent with foil; let stand for 10 minutes before carving.
Skim fat from pan juices. Whisk flour into juices; cook over medium-high heat, whisking, for 1 minute. Whisk in chicken stock; reduce heat and simmer, whisking, until thickened, about 5 minutes.
Strain into warmed gravy boat; serve with chicken.