Add the basil, pine nuts, garlic, nutritional yeast, lemon juice, and salt to a food processor. With the motor running, stream in the olive oil and blend until smooth.
Cook the penne in salted water according to package directions. Reserve 1 cup of the pasta water, then drain.
Slice the chicken breasts in half horizontally and season with salt and pepper. Cook in the olive oil in a large skillet over medium-high heat for 5-6 minutes per side until golden and 165°F, then rest and slice.
Reduce the heat to medium and melt the dairy free butter in the same skillet. Add the cherry tomatoes and cook for 2 minutes until just blistered, then sprinkle in the flour and stir for 30 seconds.
Slowly whisk in the oat milk and simmer for 2-3 minutes until it thickens. Add the spinach and stir until wilted.
Remove the skillet from the heat and stir in the pesto, which keeps the basil bright green. Add the penne and toss, adding splashes of pasta water until the sauce is silky.