Bring a medium pot of well-salted water to a boil. Toss the beef with 2 tsp. soy sauce and 1/2 tsp. salt.
Finely grate the orange zest and then juice the orange. Put the ginger and garlic in a food processor and pulse until minced. Transfer all but 1 tsp. of the mixture to a small ramekin. Add the peanut butter, zest, 2 Tbs. of the juice, and 2 Tbs. of the soy sauce to the food processor. Process until smooth, adding 1 to 2 Tbs. water so the mixture loosens a bit.
In a small bowl, mix 2 Tbs. of the remaining orange juice with the oyster sauce, the remaining 1 Tbs. soy sauce, and 2 Tbs. water.
Cook the soba noodles in the boiling water, stirring occasionally, until just tender, about 5 minutes. Drain well. Transfer the noodles to a large bowl and toss with the peanut butter mixture and half of the mint.
Meanwhile, in a large skillet, heat 1-1/2 Tbs. of the oil with the minced garlic and ginger over medium-high heat until sizzling. Add the beef and cook, stirring, until it just loses its raw color, 1-1/2 to 2 minutes. Transfer to a large plate. Reduce the heat to medium, add the remaining 1-1/2 Tbs. oil to the skillet, and then the broccoli. Cook, stirring, until browned in places, 1 to 2 minutes. Add the oyster sauce mixture, cover, and cook until the broccoli is crisp-tender, about 2 minutes. Return the beef and its juices to the pan and cook, stirring, until heated through, about 1 minute. Portion the noodles among 4 plates and top with the beef and broccoli. Sprinkle with the remaining mint and serve.