Put potatoes in a steamer basket over simmering water, cover and steam for 7 minutes. Add green beans and steam for 3 minutes more until both are partly tender, then spread on a tray to stop steaming.
Toast walnuts in a dry skillet over medium heat for 3 to 4 minutes and cool briefly. Blend with basil, garlic, olive oil, Parmesan and 1/4 teaspoon salt to make a thick pesto.
Melt sauce butter in a saucepan and stir in flour for 1 minute. Gradually whisk in milk and broth, add remaining salt and pepper and simmer for 4 to 5 minutes until lightly thickened; remove from heat and stir in all the pesto.
Butter a 6-quart slow cooker and spread 3/4 cup pesto sauce over the bottom. Divide dry noodles into three groups, breaking to fit without thick overlaps.
Layer one group of noodles, one-third of the potatoes and beans, one-third of the remaining sauce and one-third of the mozzarella. Repeat twice, keeping every noodle edge coated.
Cover and cook on low for 3 to 3 1/2 hours, checking at 2 1/2 hours, until the center pasta and potatoes are tender and the filling reaches 165°F. Switch off, uncover and rest for 20 minutes before cutting into eight portions.