How to Make It
Make the tuna confit: In a medium bowl, stir together salt and brown sugar. Add tuna; rub with salt mixture to completely coat fish. Cover and cure in refrigerator 1 hour. Rinse tuna thoroughly, and pat dry.
In a small saucepan, heat 1 cup oil, thyme, and garlic clove over low heat to 200°F. Carefully submerge tuna in oil and cook until cooked through, about 40 minutes. Using tongs, transfer tuna to paper towels to drain. Let cool completely; flake into large chunks.
Make the bagna cauda: In a small saucepan, cook 6 tablespoons oil and garlic paste over low heat, stirring, until very fragrant but not browned, about 10 minutes. Add wine and cook until reduced by half, 5 to 7 minutes. Add anchovy paste and cook, stirring, 2 minutes. Whisk in butter, 1 tablespoon at a time, until emulsified. Whisk in 1 tablespoon lemon juice and season with pepper. Keep warm over very low heat, whisking occasionally.
Make the pasta: In a large pot of salted boiling water, cook pasta until al dente. Drain; reserve 1 cup cooking liquid.
Wipe out pot, add 1/4 cup oil, and heat over moderate heat. Add red pepper and cook, stirring, until fragrant, about 1 minute. Add broccoli rabe and cook, stirring occasionally, until crisp-tender, 3 to 5 minutes. Add pasta, tuna, 2 tablespoons lemon juice, and half of reserved cooking liquid and cook, stirring gently, until hot and pasta is lightly coated, about 3 minutes, adding additional cooking liquid if needed. Season with salt and pepper.
Transfer pasta to 4 shallow bowls and drizzle with a little of the bagna cauda. Garnish with cheese and potato sticks.