Pat chicken dry; add paprika, pepper, and 1/2 teaspoon salt. Heat oil in 5-quart Dutch oven over medium-high; brown one side 5 minutes. Plate partly raw chicken.
Reduce to medium; cook onion and carrot 4 minutes. Add garlic 30 seconds, then rice and remaining salt 1 minute.
Add broth; scrape browned bits and simmer. Nestle chicken browned-side up on rice; add juices.
Cover tightly; cook over low 18 minutes without stirring. Scatter peas around chicken; cover and cook 5 to 7 minutes more until rice tender and thickest thigh 165°F.
Remove from heat; rest covered 5 minutes. Gently fluff around thighs; serve 1 thigh with one-quarter rice and vegetables per portion.