Place peeled and cubed Yukon Gold potatoes in a large pot, cover with cold salted water, and boil until fork-tender, about 15-18 minutes. Drain well and let steam escape.
Heat olive oil in a large Dutch oven over medium heat. Cook onion, carrots, and celery until softened, about 6-8 minutes. Add garlic and cook 30 seconds until fragrant.
Add chopped mushrooms to the pot and increase heat to medium-high. Cook 8-10 minutes, stirring occasionally, until mushrooms release their liquid and it evaporates, leaving them browned.
Stir tomato paste into the vegetable mixture and cook 1-2 minutes to caramelize. Sprinkle flour over the mixture and cook 1 minute, stirring to coat evenly.
Pour in vegetable broth, scraping up browned bits from the bottom. Add lentils, Worcestershire sauce, thyme, and rosemary. Bring to a boil, reduce to simmer, cover partially, and cook 25-30 minutes until lentils are tender and mixture thickens.
Stir in frozen peas during the last 3 minutes of cooking. Season with salt and pepper to taste. Remove from heat when filling is thick enough to hold its shape.
Preheat oven to 400°F. Mash drained potatoes with butter and warm milk until smooth. Stir in shredded cheddar until melted, then season with salt and pepper.
Spoon lentil filling into a 9x13 inch baking dish. Dollop mashed potatoes on top and spread gently to cover the filling completely, sealing edges against the dish. Create ridges with a fork for crispiness.
Place assembled pie on a baking sheet and bake at 400°F for 25-30 minutes until filling is bubbling and potato topping is golden brown. Rest 10 minutes before serving.