Heat oven to 450 degrees. Combine garlic, Aleppo pepper, olive oil and 2 tablespoons lemon juice in a small bowl and season with salt and pepper.
Lay the trout down flat, skin-side down, on a rimmed sheet pan, positioning the two pieces snugly next to each other. Place lemon slices on top of the trout and scatter broccolini and capers around it. Drizzle everything with olive oil and season with salt and pepper.
Place in oven and roast until broccolini has started to lightly char around the edges and the trout is opaque and cooked through, 8 to 9 minutes. Remove from oven and immediately drizzle everything with the olive oil-garlic mixture and let the flavors mingle for a minute or two.
Serve directly from the sheet, or divide fillets among 4 smaller plates