Coat a 9-1/2-inch deep-dish pie plate with butter and set aside.
Combine flour, sugar, and salt in the bowl of a food processor fitted with a blade attachment and pulse 5 times to evenly incorporate. Add butter and pulse until just incorporated (some large pieces will remain). Drizzle in 4 tablespoons ice water and pulse until moistened throughout, about 5 pulses. (Add the last tablespoon of ice water if necessary, but do not overwork the dough or it will become tough.)
Turn into the prepared pie plate. Press with floured fingers or the back of a measuring cup dipped in flour until dough evenly covers the bottom and sides of the pie plate. Place in the refrigerator to chill, at least 20 minutes.
Meanwhile, heat the oven to 400°F, arrange a rack in the lower third, and place a baking sheet lined with aluminum foil on the rack. When the oven is hot, pierce the base of the crust with a fork several times, line it with a piece of parchment paper, and fill it with pie weights or dried beans.
Place crust on the hot baking sheet and bake until dry to the touch but not browned, about 20 to 25 minutes. Remove from the oven, set aside the pie weights and parchment paper, and let crust cool slightly on a wire rack. Return the baking sheet to the oven.
For the filling:
Combine peaches, sugars, vanilla seeds, and lemon juice in a large nonreactive bowl and gently mix. Add raspberries and cornstarch and toss to coat. Transfer filling to crust and place pie on the hot baking sheet. Bake until peaches are just beginning to brown, about 30 minutes.
For the streusel:
Meanwhile, combine all streusel ingredients except butter in a medium bowl and mix until evenly incorporated. Add butter and combine with your fingers, rubbing the butter into the flour mixture until mostly incorporated but some small pieces remain. Place in the refrigerator until ready to use.
When the pie is ready, remove it from the oven and reduce the temperature to 375°F Sprinkle streusel over top and return the pie to the oven. Bake until the filling is bubbly, the streusel is brown, and the peaches are fork-tender, about 40 to 45 minutes more. Let cool on a wire rack at least 45 minutes before slicing.