Meanwhile, in a medium bowl, whisk sour cream and sriracha until smooth. If needed, add water, a few drops a time, until sauce is a good consistency for drizzling. Refrigerate until ready to use.
In another medium bowl, whisk brown sugar, soy sauce, cornstarch, vinegar, sesame oil, and ½ cup water until smooth. Let sit at room temperature until ready to use.
Remove dark green parts from scallions and thinly slice on a diagonal; set aside for serving. Thinly slice white and light green parts.
In a large nonstick skillet or wok over high heat, heat vegetable oil. Cook garlic, ginger, and white and light green scallion parts, stirring frequently, until light golden, about 1 minute. Add beef and cook, breaking up into smaller pieces, until just cooked through and starting to brown, about 5 minutes.
Add broccoli and cook, stirring frequently, until bright-green, about 2 minutes. Add carrots and reserved soy sauce mixture. Cook, stirring frequently, until sauce thickens and vegetables are crisp-tender, 3 to 4 minutes more.
Divide rice among bowls. Spoon beef mixture over top. Drizzle with sriracha-sour cream. Sprinkle with reserved dark green scallion parts and sesame seeds.
