Preheat oven to 325°. Crumble cookies into a food processor and pulse until finely ground. In a medium bowl, toss 2 tbsp. lemon zest with 1 1/2 tbsp. sugar. Add cookie crumbs and mix well. Butter sides of an 8-in. pan with removable rim and press cookie mixture into bottom; bake 15 minutes. Remove from oven and let cool in pan.
Lower temperature to 275°. In standing mixer fitted with paddle attachment, beat goat cheese and cream cheese until light and fluffy. Add remaining lemon zest, 1 tbsp. lemon juice, the orange zest, 3/4 cup sugar, the sour cream, and the vanilla; beat well. Scrape bowl and add eggs; beat well.
Pour batter onto crust and bake about 1 hour, or until set on the sides but still quite jiggly in the center and only slightly sticky. Turn oven off, crack oven door, and let cake cool in oven 2 hours. Remove and let cool completely; then chill, covered, at least 2 hours and up to overnight.
Run a thin knife between cake and rim of pan and remove rim. Thinly slice strawberries, then toss with remaining 2 1/2 tbsp. sugar and lemon juice. Arrange on cake.