Adjust oven rack to bottom position and preheat oven to 425℉ (220℃).

In a large pot, add potatoes, water, 4 teaspoons salt, and baking soda, and heat over high heat until boiling. Add potatoes and cook, lowering heat to maintain simmer, until potatoes are tender when pierced with a fork or knife, 12 to 16 minutes. Drain potatoes then return to pot and heat over medium heat, gently tossing occasionally, until potatoes are dry, about 1 minute. Remove from heat and set aside for 5 minutes.

While potatoes cool, in a large skillet, heat 1 tablespoon oil and 1 tablespoon butter over medium-high heat until butter is melted and no longer foams, about 2 minutes. In a 9-inch tart pan with a removable bottom or springform pan, use a spatula to scrape melted butter and oil into pan and use a pastry brush to brush evenly on the bottom and sides of pan (see notes). Add warm potatoes in an even layer. Use the back of a greased measuring cup or bottom of a mug to smash potatoes into an even layer, being sure to press potatoes up the sides of tart pan; make sure there are no gaps in the potato layer (use your fingers if needed). Transfer tart pan to a baking sheet. Brush 1 tablespoon oil evenly over top of potato crust and sprinkle with ½ teaspoon salt. Bake until golden brown and slightly crispy, about 40 minutes.

Meanwhile, in now-empty large skillet, heat remaining 1 tablespoon oil and butter over medium heat until butter is melted. Add leeks and cook, stirring occasionally and adjusting heat as needed to prevent scorching, until very soft and just starting to brown, about 10 minutes.

Push leeks to edge of skillet and add asparagus, thyme, garlic, black pepper, and remaining ½ teaspoon salt to center of skillet. Cook, occasionally stirring asparagus mixture in the center of skillet, until asparagus is just tender, 2 to 4 minutes. Stir asparagus into leeks until all vegetable are well combined. Remove from heat and set aside.

In a medium bowl, whisk together eggs, cream cheese, and heavy cream until smooth. Set aside until potatoes are finished cooking.

When potato crust has finished cooking, remove from oven and reduce oven temperature to 400℉ (205℃). Sprinkle fontina cheese evenly over par-cooked potato crust. Spread cooked leek mixture evenly over fontina.

Sprinkle goat cheese, parsley, and chives evenly over vegetables, then pour the egg mixture on top. Gently shake tart pan to ensure egg mixture evenly covers vegetables. Bake until eggs are just cooked through and set, 16 to 20 minutes (center will still be slightly soft and jiggly). Transfer tart pan to a wire rack and let cool for 10 minutes. Run a sharp knife along edges of tart pan to loosen potato crust away from pan. Carefully remove the edge of the tart pan. Before serving, sprinkle with parsley and chives. Serve warm or at room temperature with sour cream or crème fraîche.