Preheat oven to 400°.
In a small bowl, cover mushrooms with hot water. Let stand for 10 minutes.
Heat a cast-iron braiser over medium-high heat. Add oil and butter. Sprinkle both sides of rabbit with salt and pepper. Add half of rabbit to pot; cook until browned, 3 to 4 minutes per side. Remove from pot, and set aside. Repeat with remaining rabbit.
Reduce heat to medium. Add onion, carrot, and celery to braiser; scrape browned bits from bottom of pot with a wooden spoon. Sauté, stirring occasionally, until softened and golden brown, 8 to 10 minutes. Stir in flour; cook until flour is absorbed by vegetables, about 1 minute. Add broth and wine; bring to a boil, stirring until liquid just thickens, about 45 seconds.
Return rabbit to pot. Add mushrooms, artichokes, tarragon, lemon juice, lemon slices, thyme, and bay leaf. Cover with lid.
Bake until a food thermometer inserted in thickest portion of rabbit registers 160°, about 20 minutes. Serve immediately.