Rub brisket all over with salt, pepper, and paprika. Let it sit at room temperature on a large rimmed baking sheet for 1 hour.
Preheat oven to 325°F. Heat oil in a large Dutch oven over medium-high. Add brisket and cook until well browned on both sides, 3 to 4 minutes per side. Return to baking sheet.
Reduce heat to medium and add yellow onions and pearl onions to Dutch oven. Cook, stirring often, until softened and beginning to caramelize, about 15 minutes.
Stir in tomato paste and brown sugar; cook, stirring constantly, until slightly caramelized, about 2 minutes. Stir in beer, water, and onion soup mix; bring to a simmer over medium.
Return brisket to Dutch oven, fat side down, nestling in the onion mixture. (Half of the brisket should be submerged in liquid; if not, add more beer.) Cover and bake in preheated oven until brisket is very tender, checking periodically to be sure brisket is always halfway submerged in the braising liquid, about 3 hours, flipping brisket after 1 1/2 hours.
Remove from oven and let rest at room temperature before carving, about 20 minutes. While the brisket rests, strain the onions, and pour the reserved braising liquid into a deep saucepan. Bring it to a simmer over medium-high heat until it’s reduced by half and thickened into a light gravy. Allow it to reduce further to your preferred consistency, then stir the onions back in and serve alongside the brisket.