Sauté Brussels Sprouts: Heat olive oil in a large pot over medium-high heat.
Sauté garlic and Brussels sprouts for 5 minutes, seasoning with salt and pepper. Add 1 tablespoon lemon juice, toss, reduce heat to medium-low, and cover. Cook for 10 minutes or until tender. Remove and set aside.
Prepare Sauce Base: In the same pot, melt vegan butter over medium-high heat.
Whisk in flour, cooking for 1-2 minutes.
Gradually whisk in vegetable broth, non-dairy milk, salt, and pepper until smooth.
Cook Spaghetti: Bring the sauce to a boil, then add spaghetti (broken in half for easier cooking).
Reduce heat to medium, cover, and simmer for 8-10 minutes, stirring occasionally to prevent sticking and ensure even cooking.
Adjust Sauce: Once the spaghetti is cooked, the liquid should thicken into a sauce. Add the remaining lemon juice, salt, and pepper. Taste and adjust seasoning as needed.
Combine and Serve: Add the cooked Brussels sprouts and garlic back into the pot. Toss to combine and heat for 1-2 minutes.
Serve hot with extra lemon juice and vegan Parmesan if desired.